Questions and Answers
If f is the function defined by f(x)=x²−9x²+2x−15, what is limx→3f(x)?
3/4
What is limx→3 (x−3)/(x³−9x)?
1/18
Given that limx→0sin(2x)/(2x)=1, what is limx→0cos(5x)/(8x cot(2x))?
1/4
What is limx→−4 (x+4)/(x³−16x)?
Signup and view all the answers
If limx→0sin(2x)/(2x)=1, what is limx→0tan(2x)/(6x sec(3x))?
Signup and view all the answers
If f is the function defined by f(x)=x²−4x²+x−6, what is limx→2f(x)?
Signup and view all the answers
Study Notes
Limits Evaluation Techniques
-
Limit of a Rational Function
For the function f(x) defined as ( f(x) = \frac{x^2 - 9}{x^2 + 2x - 15} ) at ( x \to 3 ), the limit is computed as ( \frac{3}{4} ). -
Factoring Approach
Simplifying the expression involves factoring and cancelling common terms:
( f(x) = \frac{(x+3)(x-3)}{(x+5)(x-3)} = \frac{x+3}{x+5} ).
The limit can then be resolved to ( \frac{6}{8} = \frac{3}{4} ).
Limit of Difference Quotients
-
Limit Calculation Example
For the limit of ( \frac{x - 3}{x^3 - 9x} ) as ( x \to 3 ), the result is ( \frac{1}{18} ). -
Rearranging the Expression
The expression can be factorized:
( x^3 - 9x = x(x+3)(x-3) ).
The limit simplifies to ( \lim_{x \to 3} \frac{1}{x(x+3)} = \frac{1}{1 \cdot 6} = \frac{1}{18} ).
Trigonometric Limits
-
Calculating Limit with Sine
Utilizing known limits, ( \lim_{x \to 0} \frac{\sin(2x)}{2x} = 1 ) aids in finding ( \lim_{x \to 0} \frac{\cos(5x)}{8x \cot(2x)} ).
This limit simplifies to ( \frac{1}{4} ). -
Using Cotangent Form
The cotangent can be expressed in terms of sine and cosine, allowing manipulation of the limit to achieve the final answer of ( \frac{1}{4} ).
Further Limit Evaluations
-
Limit at Negative Values
For ( \lim_{x \to -4} \frac{x + 4}{x^3 - 16x} ), the limit evaluates to ( \frac{1}{32} ). -
Factoring and Simplifying
Factoring results in:
( x^3 - 16x = x(x - 4)(x + 4) ).
The limit computation then concludes as ( \frac{1}{1 \cdot (-8)} = \frac{1}{32} ).
Combining Functions
-
Tan and Secant Limit
To find ( \lim_{x \to 0} \frac{\tan(2x)}{6x \sec(3x)} ), the result is ( \frac{1}{3} ). -
Rewriting Using Trigonometric Identities
By expressing the tangent and secant functions in terms of sine and cosine, the limit simplifies to achieve the value of ( \frac{1}{3} ).
Evaluating Limits for Specific Functions
-
Limit at a Point
For ( f(x) = \frac{x^2 - 4}{x^2 + x - 6} ) at ( x \to 2 ), the limit is ( \frac{4}{5} ). -
Factoring Methodology
After factoring ( f(x) = \frac{(x-2)(x+2)}{(x-2)(x+3)} ), the limit resolves as:
( \frac{2+2}{2+3} = \frac{4}{5} ) as ( x ) approaches 2.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of limits using algebraic manipulation with this quiz. It covers various limit evaluations and approaches to solving function limits. Get ready to sharpen your calculus skills!
