Evaluating Limits: Graphs, Equations, Properties

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Questions and Answers

Explain how the Squeeze Theorem can be used to evaluate limits. Provide a general example (without specific functions) to illustrate your explanation.

The Squeeze Theorem states that if $g(x) \leq f(x) \leq h(x)$ for all $x$ in an interval containing $c$ (except possibly at $c$) and $\lim_{x \to c} g(x) = L = \lim_{x \to c} h(x)$, then $\lim_{x \to c} f(x) = L$. Essentially, $f(x)$ is 'squeezed' between two functions that approach the same limit.

Describe how the concept of a limit is used to define the derivative of a function. Provide the limit definition of the derivative.

The derivative of a function $f(x)$ is defined as the limit of the difference quotient as $h$ approaches zero. This limit represents the instantaneous rate of change of the function at a point. The limit definition of the derivative is: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.

Consider the function $f(x) = \frac{x^2 - 4}{x - 2}$. While this function is not defined at $x = 2$, evaluate $\lim_{x \to 2} f(x)$ and explain whether the discontinuity at $x = 2$ is removable or non-removable. Justify your answer.

$\lim_{x \to 2} f(x) = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} (x+2) = 4$. The discontinuity at $x = 2$ is removable because the limit exists at that point.

Explain the difference between average velocity and instantaneous velocity. How is each calculated, and what does each represent?

<p>Average velocity is the change in displacement over a time interval ($\frac{\Delta x}{\Delta t}$), representing the constant speed needed to cover the same distance in that time. Instantaneous velocity is the limit of average velocity as the time interval approaches zero ($\lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$), representing the velocity at a specific moment.</p> Signup and view all the answers

Given a piecewise function $$f(x) = \begin{cases} x^2, & x < 1 \ ax + b, & 1 \leq x < 3 \ 4, & x \geq 3 \end{cases}$$. Determine the values of $a$ and $b$ that make $f(x)$ continuous everywhere.

<p>For continuity at $x=1$, $1^2 = a(1) + b$, so $a + b = 1$. For continuity at $x=3$, $3a + b = 4$. Solving this system of equations gives $a = \frac{3}{2}$ and $b = -\frac{1}{2}$.</p> Signup and view all the answers

Flashcards

What is a limit?

A value that a function approaches as the input approaches some value.

How to evaluate limits from a graph?

Examine the behavior of the curve as you get closer and closer to the x value from both sides.

What is a continuous function?

A function is continuous if you can draw its graph without lifting your pen; there are no breaks, jumps, or holes.

What is Tangent Slope?

The slope of the line tangent to the curve at a specific point on the curve.

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Average vs. Instantaneous Velocity?

Average velocity is the change in position over a time interval; instantaneous velocity is the velocity at a specific moment.

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Study Notes

Limits can be evaluated from graphs, equations, or by using properties of limits.

Graph Analysis for Limits

To evaluate limits from a graph, observe the y-value the function approaches as x approaches a specific point from both sides.

Equation Analysis for Limits

To evaluate limits from an equation, substitute the x-value into the function if it is continuous at that point.
If direct substitution results in an indeterminate form (e.g., 0/0), algebraic manipulation or L'Hôpital's Rule may be necessary.

Properties of Limits

Limits can be determined using various properties, such as the sum, product, quotient, and power rules.

Continuity and Discontinuity

Continuity and discontinuity of a graph refers to whether a function is without any interruptions such as holes or asymptotes, and requires that the limit exists at that point, the function is defined at that point, and the limit equals the function value.
A graph can be drawn as piece-wise or based on given properties.

Tangent Lines

The slope of a tangent line at a specific point is the derivative of the function evaluated at that point.
Equation of a tangent is derived using the point-slope form, utilizing the derivative at the point of tangency to find the slope.

Velocity

Average velocity is the change in displacement over the change in time over a time interval.
Instantaneous velocity is the velocity at a single point in time.

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Description

Explore how to evaluate limits using graphs, equations, and limit properties. Learn to analyze function behavior as x approaches a certain point, including the use of direct substitution and algebraic manipulation. Understand continuity, discontinuity, piece-wise functions and asymptotes.