Basic Calculus - Q3 Module 2 PDF

# **BASIC CALCULUS** - Quarter 3 - Module 2 - Limits of Transcendental Functions and Special Limits ## **Development Team of the Module** - **Writer:** Littie Beth S. Bernadez - **Editor:** Gil S. Dael - **Reviewer:** Ronald G. Tolentino - **Layout Artist:** Radhiya A. Ababon - **Management Team:...

# **BASIC CALCULUS** - Quarter 3 - Module 2 - Limits of Transcendental Functions and Special Limits ## **Development Team of the Module** - **Writer:** Littie Beth S. Bernadez - **Editor:** Gil S. Dael - **Reviewer:** Ronald G. Tolentino - **Layout Artist:** Radhiya A. Ababon - **Management Team:** - Senen Priscillo P. Paulin CESO V Elisa L. Baguio - EdD Joelyza M. Arcilla EdD, CESE Rosela R. Abiera - Marcelo K. Palispis JD, EdD Maricel S. Rasid - Nilita L. Ragay EdD Elmar L. Cabrera ## **Introductory Message** This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson. Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you. Pre-tests are provided to measure your prior knowledge on lessons in each SLM. 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Thank you. ## **Limits of Transcendental Functions (Exponential, Logarithmic, Trigonometric Functions) & Special Limits** ### **Prior Knowledge** #### **Limit Laws** Suppose that c is a constant and the limits | | | | ------ | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | 1. | lim [ **→** **→** **(x)**] = lim**→** **→** **(x)** + lim**→** **→** **(x)** and lim**→** **→** **(x)** exist, then lim**→** **→** [**→** **→** **(x)** + **→** **→** **(x)**] = lim**→** **→** **(x)** + lim**→** **→** **(x)**. | | 2. | lim [**→** **→** **(x)**] = lim**→** **→** **(x)** – lim**→** **→** **(x)** and lim**→** **→** **(x)** exist, then lim**→** **→** [**→** **→** **(x)** – **→** **→** **(x)**] = lim**→** **→** **(x)** – lim**→** **→** **(x)**. | | 3. | lim [**→** **→** **(x)**] = c*lim**→** **→** **(x)** , if c is a constant and lim**→** **→** **(x)** is finite | | 4. | lim [**→** **→** **(x)**] = lim**→** **→** **(x)** * lim**→** **→** **(x)** , if lim**→** **→** **(x)** and lim**→** **→** **(x)** are finite | | 5. | lim**→** **→** c = c | | 6. | lim**→** **→** [**→** **→** **(x)**]**n** = [lim**→** **→** **(x)**]**n**, if lim**→** **→** **(x)** ≠ 0 , where n is a positive integer | | 7. | lim**→** **→** **→** **→** **(x)** = **→** **→** **(x)**, if lim**→** **→** **(x)** exists | | 8. | lim**→** **→** [**→** **→** **(x)**]**n** = 1, if lim**→** **→** **(x)** = 1, where n is a positive integer | | 9. | lim**→** **→** **→** **→** **(x)** = **→** **→** **(x)**, where n is a positive integer | | 10. | lim**→** **→** [**→** **→** **(x)**]**n** = **→** **→** **(x)**, where n is a positive integer , if lim**→** **→** **(x)** exists | | 11. | lim**→** **→** **→** **→** **(x)** / **→** **→** **(x)** = lim**→** **→** **→** **→** **(x)** / lim**→** **→** **(x)**, where n is a positive integer | ### **Pre-Assessment** Evaluate the following limits by constructing table of values: 1. lim**→** 13 2. lim**→** -25 ### **Evaluating Limits of Exponential Function** Consider the natural exponential function **→** **→** **(x)** = **→** **→** **x**, where **→** **→** is called the Euler number, and has the value of 2.718281.... #### **Illustrative Example 1** Evaluate the lim **→** 0 **→** **→** **x** **Solution:** There are two means to evaluate the limit of the function: through a table of values and a graphical approach. **Through Tables of Values:** Construct the table of values for **→** **→** **x**. We will do so by approaching the number 0 from the left or through the values less than but close to 0, and by approaching the number 0 from the right or through the values greater than but close to 0. | | **→** **→** **x** = **→** **→** **→****→** | | -------------- | ------------------------------------- | | **→** **→** = -1 | 0.36787944117 | | **→** **→** = -0.5 | 0.60653065971 | | **→** **→** = -0.1 | 0.90483741803 | | **→** **→** = -0.01 | 0.99004983374 | | **→** **→** = -0.001 | 0.99900049983 | | **→** **→** = -0.0001 | 0.999900049983 | | **→** **→** = -0.00001 | 0.99999000005 | **Approaching 0 from the Right** | | **→** **→** **x** = **→** **→** **→** **→** | | -------------- | ------------------------------------- | | **→** **→** = 1 | 2.71828182846 | | **→** **→** = 0.5 | 1.6487212707 | | **→** **→** = 0.1 | 1.10517091808 | | **→** **→** = 0.01 | 1.01005016708 | | **→** **→** = 0.001 | 1.00100050017 | | **→** **→** = 0.0001 | 1.000100005 | | **→** **→** = 0.00001 | 1.00001000005 | **Through a Graph** [insert graph here] Therefore, the limit of **→** **→** **x** = 1 as **→** **→** approaches to 0 from the right and the left. ### **Evaluating Limits of Logarithmic Function** Consider the natural logarithmic function **→** **→** **(x)** = ln **→** **→** **x**. Recall that ln **→** **→** **x** = log **→** **→** **x**. Moreover, ln **→** **→** **x** is the inverse of the natural exponential function **→** **→** **x** or $e^x$. #### **Illustrative Example 2** Evaluate the lim **→** **→** 1 ln **→** **→** **x** **Solution** We will evaluate the limit through constructing the tables of values. Construct the table of values for **→** **→** **(x)** = ln **→** **→** **x**. We start by approaching the number 1 from the left or through the values less than but close to 1, and then approach the number 1 from the right or through the values greater than but close to 1. **Approaching 1 from the left** | | **→** **→** **(x)** = ln **→** **→** **x** | | -------------- | ------------------------------------- | | **→** **→** = 0.1 | -2.30258509299 | | **→** **→** = 0.5 | -0.69314718056 | | **→** **→** = 0.9 | -0.10536051565 | | **→** **→** = 0.99 | -0.01005033585 | | **→** **→** = 0.999 | -0.00100050033 | | **→** **→** = 0.9999 | -0.0001000005 | | **→** **→** = 0.99999 | -0.00001000005 | **Approaching 1 from the right** | | **→** **→** **(x)** = ln **→** **→** **x** | | -------------- | ------------------------------------- | | **→** **→** = 2 | 0.69314718056 | | **→** **→** = 1.5 | 0.4054651081 | | **→** **→** = 1.1 | 0.0953101798 | | **→** **→** = 1.01 | 0.00995033085 | | **→** **→** = 1.001 | 0.00099950033 | | **→** **→** = 1.0001 | 0.000099995 | | **→** **→** = 1.00001 | 0.00000999995 | Therefore, the limit of **→** **→** **x** = ln **→** **→** **x** = 0 as **→** **→** approaches to 1 from the left and the right. #### **Illustrative Example 3** Evaluate the lim **→** **→** 1 log **→** **→** **x** **Solution:** Since, by common logarithmic function **→** **→** **(x)** = log10 **→** **→** **x**, log**→** **→** **x** = log10**→** **→** **x**. Using tables of values, we can determine the lim**→** **→** 1 log**→** **→** **x**. Through tables of values: We shall construct the table of values for **→** **→** **(x)** = log **→** **→** **x**. **Approaching 1 from the left** | | **→** **→** **(x)** = log **→** **→** **x** | | -------------- | ------------------------------------- | | **→** **→** = 0.1 | -1 | | **→** **→** = 0.5 | -0.30102999566 | | **→** **→** = 0.9 | -0.04575749056 | | **→** **→** = 0.99 | -0.0043648054 | | **→** **→** = 0.999 | -0.00043451177 | | **→** **→** = 0.9999 | 0.00004343161 | | **→** **→** = 0.99999 | 0.00000434296 | **Approaching 1 from the right** | | **→** **→** **(x)** = log **→** **→** **x** | | -------------- | ------------------------------------- | | **→** **→** = 2 | 0.3010299956 | | **→** **→** = 1.5 | 0.17609125905 | | **→** **→** = 1.1 | 0.04139268515 | | **→** **→** = 1.01 | 0.00432137378 | | **→** **→** = 1.001 | 0.00043407747 | | **→** **→** = 1.0001 | 0.00004342727 | | **→** **→** = 1.00001 | 0.00000434292 | Therefore the limit of **→** **→** **x** = log **→** **→** **x** = 0 as **→** **→** approaches to 1 from the left and the right. #### **Illustrative Example 4** Evaluate the lim **→** **→** 1 ln **→** **→** **x** and lim**→** **→** 1 log**→** **→** **x** through a graph. **Solution:** The graphs of both the natural and common logarithmic functions can be used to determine the limits as **→** **→** approaches to 1. [insert graph here] Therefore, lim **→** **→** 1 ln **→** **→** **x** and lim**→** **→** 1 log**→** **→** **x** = 0. ### **Evaluating Limits of Trigonometric Functions** #### **Illustrative Example 5** Evaluate the lim**→** **→** 0 sin **→** **→** **x** **Solution:** We will evaluate the lim**→** **→** 0 sin **→** **→** **x** by using tables of values and graphical approach. Construct the table of values for **→** **→** **(x)** = sin **→** **→** **x**. We start by approaching the number 0 from the left or through the values less than but close to 0, and approaching the number 0 from the right or through the values greater than but close to 0. **Approaching 0 from the left** | | **→** **→** **(x)** = sin **→** **→** **x** | | -------------- | ------------------------------------- | | **→** **→** = -1 | -0.8414709848 | | **→** **→** = -0.5 | -0.4794255386 | | **→** **→** = -0.1 | -0.09983341664 | | **→** **→** = -0.01 | -0.00999983333 | | **→** **→** = -0.001 | -0.00099999983 | | **→** **→** = -0.0001 | -0.00009999999 | | **→** **→** = -0.00001 | -0.00000999999 | **Approaching 0 from the right** | | **→** **→** **(x)** = sin **→** **→** **x** | | -------------- | ------------------------------------- | | **→** **→** = 1 | 0.8414709848 | | **→** **→** = 0.5 | 0.4794255386 | | **→** **→** = 0.1 | 0.09983341664 | | **→** **→** = 0.01 | 0.00999983333 | | **→** **→** = 0.001 | 0.00099999983 | | **→** **→** = 0.0001 | 0.00009999999 | | **→** **→** = 0.00001 | 0.00000999999 | Therefore, the limit of **→** **→** **x** = sin **→** **→** **x** = 0 as **→** **→** approaches to 0 from the right and the left. **Through a graph:** We can also find the lim**→** **→** 0 sin **→** **→** **x** by using the graph of the sine function. [insert graph here] By inspecting, as the value of **→** **→** approaches to zero from either left or right of it, **→** **→** **(x)** approaches to 0. Thus, lim**→** **→** 0 sin **→** **→** **x** = 0 ### **Evaluating Special Limits** #### **Illustrative Example 1** Evaluate the lim**→** **→** 0 sin**→** **→** **x** / **→** **→** **x** **Solution** Using tables of values and graphical approach, we construct the table of values for **→** **→** **(x)** = sin**→** **→** **x** / **→** **→** **x**. **Through tables of values:** We start by approaching the number 0 from the left or through the values less than but close to 0, and approaching the number 0 from the right or through the values greater than but close to 0. **Approaching 0 from the left** | | **→** **→** **(x)** = sin **→** **→** **x** / **→** **→** **x** | | -------------- | --------------------------------------------------- | | **→** **→** = -1 | 0.8414709848 | | **→** **→** = -0.5 | 0.9588510772 | | **→** **→** = -0.1 | 0.9983341665 | | **→** **→** = -0.01 | 0.9999833334 | | **→** **→** = -0.001 | 0.99999998333 | | **→** **→** = -0.0001 | 0.99999999983 | **Approaching 0 from the right** | | **→** **→** **(x)** = sin **→** **→** **x** / **→** **→** **x** | | -------------- | --------------------------------------------------- | | **→** **→** = 1 | 0.8414709848 | | **→** **→** = 0.5 | 0.9588510772 | | **→** **→** = 0.1 | 0.9983341665 | | **→** **→** = 0.01 | 0.9999833334 | | **→** **→** = 0.001 | 0.99999998333 | | **→** **→** = 0.0001 | 0.99999999983 | Therefore, the limit of **→** **→** **x** = sin **→** **→** **x** / **→** **→** **x** = 1 as **→** **→** approaches to 0 from the right and the left. Since, lim**→** **→** 0- sin**→** **→** **x** / **→** **→** **x** = 1 and lim**→** **→** 0+ sin**→** **→** **x** / **→** **→** **x** = 1 are both equal to 1. Then, lim**→** **→** 0 sin**→** **→** **x** / **→** **→** **x** = 1. **Through a graph:** [insert graph there] The graph of **→** **→** **(x)** = sin**→** **→** **x** / **→** **→** **x** below confirms that the **→** **→** **(x)** values approach 1 as **→** **→** approaches to 0. #### **Illustrative Example 6** Evaluate the lim**→** **→** 0 1 - cos**→** **→** **x** **Solution** Using the tables of values and the graph of the function, we will evaluate lim**→** **→** 0 1 - cos**→** **→** **x**. **Through tables of values:** Construct the table of values for **→** **→** **(x)** = 1 - cos**→** **→** **x**. We start by approaching the number 0 from the left or through the values less than but close to 0 and from the right or through the values greater than but close to 0. **Approaching 0 from the left** | | **→** **→** **(x)** = 1 - cos**→** **→** **x** | | -------------- | ------------------------------------------- | | **→** **→** = -1 | -0.4596976941 | | **→** **→** = -0.5 | -0.2448348762 | | **→** **→** = -0.1 | -0.04995834722 | | **→** **→** = -0.01 | -0.0049999583 | | **→** **→** = -0.001 | -0.0004999999 | | **→** **→** = -0.0001 | -0.00005 | **Approaching 0 from the right** | | **→** **→** **(x)** = 1 - cos**→** **→** **x** | | -------------- | ------------------------------------------- | | **→** **→** = 1 | 0.4596976941 | | **→** **→** = 0.5 | 0.2448348762 | | **→** **→** = 0.1 | 0.04995834722 | | **→** **→** = 0.01 | 0.0049999583 | | **→** **→** = 0.001 | 0.0004999999 | | **→** **→** = 0.0001 | 0.00005 | Therefore the limit of **→** **→** **x** = 1 - cos**→** **→** **x** = 0 as **→** **→** approaches to 0 from the left and the right. Since, lim**→** **→** 0- 1 - cos**→** **→** **x** = 0 and lim**→** **→** 0+ 1 - cos**→** **→** **x** = 0 are both equal to 0. Then, lim**→** **→** 0 1 - cos**→** **→** **x** = 0. **Through a graph:** [insert graph there] The graph of **→** **→** **(x)** = 1 - cos**→** **→** **x** below confirms that the **→** **→** **(x)** values approach 0 as **→** **→** approaches to 0. #### **Illustrative Example 7** Evaluate the lim**→** **→** 0 (**→** **→** **x** - 1)/**→** **→** **x** **Solution** As with the other examples, we shall evaluate lim**→** **→** 0 (**→** **→** **x** - 1)/**→** **→** **x** through tables of values. Construct the table of values for **→** **→** **(x)** = (**→** **→** **x**-1)/**→** **→** **x**. We start by approaching the number 0 from the left or through the values less than but close to 0, and approaching the number 0 from the right or through the values greater than but close to 0. **Approaching 0 from the left** | | **→** **→** **(x)** = (**→** **→** **x** - 1)/**→** **→** **x** | | -------------- | ------------------------------------------------------ | | **→** **→** = -1 | 0.6321205588 | | **→** **→** = -0.5 | 0.7869386806 | | **→** **→** = -0.1 | 0.9516258196 | | **→** **→** = -0.01 | 0.9950166251 | | **→** **→** = -0.001 | 0.9995001666 | | **→** **→** = -0.0001 | 0.9999500016 | **Approaching 0 from the right** | | **→** **→** **(x)** = (**→** **→** **x** - 1)/**→** **→** **x** | | -------------- | ------------------------------------------------------ | | **→** **→** = 1 | 1.718281828 | | **→** **→** = 0.5 | 1.297442541 | | **→** **→** = 0.1 | 1.051709181 | | **→** **→** = 0.01 | 1.005016708 | | **→** **→** = 0.001 | 1.000500167 | | **→** **→** = 0.0001 | 1.000050002 | Therefore the limit of **→** **→** **x** = (**→** **→** **x** - 1)/**→** **→** **x** = 1 as **→** **→** approaches to 0 from the left and the right. Since, lim**→** **→** 0- (**→** **→** **x** - 1)/**→** **→** **x** = 1 and lim**→** **→** 0+ (**→** **→** **x** - 1)/**→** **→** **x** = 1 are both equal to 1. Then, lim**→** **→** 0 (**→** **→** **x** - 1)/**→** **→** **x** = 1. **Through a graph:** [insert graph there] The graph of **→** **→** **(x)** = (**→** **→** **x** - 1)/**→** **→** **x** below confirms that the **→** **→** **(x)** values approach 1 as **→** **→** approaches to 0. ## **I Have Learned** The limit of a function and the functional value at a point is lim **→** **→** **(x)** = **→** **→** **(c)** either **→** **→** **(x)** is exponential, logarithmic or trigonometric and c is a real number which is in the domain of **→** **→** **(x)**. Special limits such as, - lim**→** **→** 0 1 - 1 = -1 - lim**→** **→** 0 sin **→** **→** **x** / **→** **→** **x** = 1 - lim**→** **→** 0 1 - cos**→** **→** **x** / **→** **→** **x** = 0 "0" upon direct substitution. However, can be found either by using tables of values and graphs of the function. ## **I Can Do** Evaluate lim**→** **→** 1 (√**→** **→** + 1 - 1)/√**→** **→** + 1. Write your solutions comprehensively. (Hint: Direct substitution will result in the expression `"0/0"`. To resolve, rationalize the given function first before applying the limit). Please be guided with the rubric. ## **Rubric** | Category | 5 | 4 | 3 | 2 | | ---------------------------- | ---------------------------------------------------------------- | -------------------------------------------------------------- | --------------------------------------------------------------------------- | ---------------------------------------------------------------- | | **Mathematical Concepts** | Explanation shows complete understanding of the mathematical concepts used to solve the problem(s). | Explanation shows substantial understanding of the mathematical concepts used to solve the problem(s). | Explanation shows some understanding of the mathematical concepts needed to solve the problem(s). | Explanation shows very limited understanding of the underlying concepts needed to solve the problem(s). | | **Mathematical Errors** | 90-100% of the steps and solutions have no mathematical errors. | Almost all (85-89%) of the steps and solutions have no mathematical errors. | Most (75-84%) of the steps and solutions have no mathematical errors. | More than 75% of the steps and solutions have mathematical errors. | | **Neatness and Organization** | The work is presented in a neat, clear, organized fashion that is easy to read. | The work is presented in a neat and organized fashion that is easy to read. | The work is presented in an organized fashion but may be hard to read at times. | The work appears sloppy and unorganized. It is hard to know what information goes together. | | **Completion** | All problems are completed. | All but one of the problems are completed. | All but two of the problems are completed. | Several of the problems are not completed. | ## **References** - Arceo, Carlene P., Lemence, Richard S. 2016. Basic Calculus Teaching Guide for Senior High School. Quezon City: Department of Education - Bureau of Learning Resources (DepEd-BLR). - Bittinger; Ellenbogen; Srgent. n.d. "Calculus and Its Applications, 11th Edition." University of Illinois Chicago. Accessed 2021. https://bit.ly/3qVgUT3. - Guichard, David, and Albert Schuler. 2021. Single Variable Calculus. San Francisco, California, USA: Creative Commons Attribution - NonCommercial ShareAlike License . - Weisstein, Eric W. n.d. "WolframMathWorld." https://mathworld.wolfram.com/. - https://mathworld.wolfram.com/TranscendentalFunction.html. ## **For Inquiries and Feedback** - Department of Education – Schools Division of Negros Oriental Kagawasan, Ave., Daro, Dumaguete City, Negros Oriental - Tel #: (035) 225 2376 /