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Chapter 1: Problem 26

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. $$ \lim _{x \rightarrow-1} \frac{x^{3}+1}{x+1} $$

Short Answer

Expert verified
The limit of the function as \(x \rightarrow-1\) is 3 and the simpler function that agrees with the given function at all points except x=-1 is \(x^2-x+1\).

Step by step solution

01

Function Simplification

Begin by factoring the numerator. The factorization of \(x^3+1\) is \((x+1)(x^2-x+1)\). So the function \(\frac{x^{3}+1}{x+1}\) becomes \((x+1)(x^2-x+1)\div(x+1)\).
02

Cancel out Common Factors

Here, \(x+1\) is a common factor in the numerator and the denominator. So we can cancel out this common factor. The function thus becomes \(x^2-x+1\). Note that our extracted simpler function is valid for all \(x\) except -1, which aligns with the exercise requirement.
03

Evaluate Limit

We can now evaluate the limit of the simplified function as \(x \rightarrow -1\). Substituting -1 into the simplified function \(x^2-x+1\), we get \((-1)^2-(-1)+1 = 1+1+1 = 3\). So, \(\lim _{x \rightarrow-1} \frac{x^{3}+1}{x+1}\) equals 3

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Most popular questions from this chapter

What is meant by an indeterminate form?

Prove or disprove: if \(x\) and \(y\) are real numbers with \(y \geq 0\) and \(y(y+1) \leq(x+1)^{2},\) then \(y(y-1) \leq x^{2}\)

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

(a) Let \(f_{1}(x)\) and \(f_{2}(x)\) be continuous on the closed interval \([a, b]\). If \(f_{1}(a)f_{2}(b),\) prove that there exists \(c\) between \(a\) and \(b\) such that \(f_{1}(c)=f_{2}(c)\). (b) Show that there exists \(c\) in \(\left[0, \frac{\pi}{2}\right]\) such that \(\cos x=x\). Use a graphing utility to approximate \(c\) to three decimal places.

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{2}+x-1, \quad[0,5], \quad f(c)=11 $$
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