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Chapter 1: Problem 25

Find the limit of the function (if it exists). Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result. $$ \lim _{x \rightarrow-1} \frac{x^{2}-1}{x+1} $$

Short Answer

Expert verified
The limit of the function as x approaches -1 is -2. The function that agrees with the original is \(f(x) = x - 1\).

Step by step solution

01

Simplifying the function

Factorize the numerator into \((x-1)(x+1)\). This leads to a new expression of the function \(f(x) = \frac{(x-1)(x+1)}{x+1}\).
02

Eliminate common factors

Notice that \(x+1\) is a common factor in the numerator and the denominator, which we can cancel out. This results in a simpler function: \(f(x) = x-1\).
03

Calculate limit

Substitute \(x = -1\) into the simplified function to obtain the limit. This gives \(\lim _{x \rightarrow-1} (x-1) = -1 -1 = -2\).
04

Confirm result using a graphing utility

A graphing utility can be used to graph the function \(f(x) = x-1\) and the original function \(f(x) = \frac{x^{2}-1}{x+1}\) tro validate the result. It can be observed that the two functions agree at all points, except at \(x = -1\), where the original function is undefined.

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Most popular questions from this chapter

True or False? In Exercises \(50-53\), determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) has a vertical asymptote at \(x=0,\) then \(f\) is undefined at \(x=0\)

Prove that if a function has an inverse function, then the inverse function is unique.

$$ \begin{aligned} &\text { Prove that if } f \text { and } g \text { are one-to-one functions, then }\\\ &(f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x). \end{aligned} $$

Explain why the function has a zero in the given interval. $$ \begin{array}{lll} \text { Function } & \text { Interval } \\ \hline f(x)=x^{2}-4 x+3 & {[2,4]} \\ \end{array} $$

Numerical and Graphical Analysis Use a graphing utility to complete the table for each function and graph each function to estimate the limit. What is the value of the limit when the power on \(x\) in the denominator is greater than \(3 ?\) $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 0.5 & 0.2 & 0.1 & 0.01 & 0.001 & 0.0001 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & & \\ \hline \end{array} $$ (a) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x}\) (b) \(\lim _{x \rightarrow 0^{-}} \frac{x-\sin x}{x^{2}}\) (c) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{3}}\) (d) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{4}}\)
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