Chapter 2

Let $f(x)=\frac{2e^x+5e^{3x}}{e^{2x}-e^{3x}}.$ Analyze $\underset{x\to 0^{-}}{lim}f(x),\underset{x\to 0^{+}}{lim}f(x),\underset{x\to -\mathrm{\infty }}{lim}f(x)$ and $\underset{x\to \mathrm{\infty }}{lim}f(x).$ Then give the horizontal and vertical asymptotes of $f.$ Plot $f$ to verify your results.

Finding a constant Suppose $$g(x)=\left\{$$\begin{array}{ll}{x}^2-5x& \text{ if }x\le -1\\ a{x}^3-7& \text{ if }x>-1\end{array}$$\right.$$ Determine a value of the constant $a$ for which $\underset{x\to -1}{lim}g(x)$ exists and state the value of the limit, if possible.

Let $f(x)=\frac{2e^x+10e^{-x}}{e^x+e^{-x}}.$ Analyze $\underset{x\to 0}{lim}f(x),\underset{x\to -\mathrm{\infty }}{lim}f(x),$ and $\underset{x\to \mathrm{\infty }}{lim}f(x).$ Then give the horizontal and vertical asymptotes of $f.$ Plot $f$ to verify your results.

Looking ahead to sequences A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence $2,4,6,8,\dots$ is specified by the function $f(n)=2n,$ where $n=1,2,3,\dots ..$ The limit of such a sequence is $\underset{n\to \mathrm{\infty }}{lim}f(n),$ provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. $\text{Missing left or extra right}$ which is defined by $f(n)=\frac{n-1}{n},$ for $n=1,2,3,\dots$

Looking ahead to sequences A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence $2,4,6,8,\dots$ is specified by the function $f(n)=2n,$ where $n=1,2,3,\dots ..$ The limit of such a sequence is $\underset{n\to \mathrm{\infty }}{lim}f(n),$ provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. $\text{Missing left or extra right}$ which is defined by $f(n)=\frac{n^2}{n+1},$ for $n=1,2,3,\dots$

Applying the Intermediate Value Theorem Use the Intermediate Value Theorem to verify that the following equations have three solutions on the given interval. Use a graphing utility to find the approximate roots. $$x^3+10x^2-100x+50=0;(-20,10)$$

Calculate the following limits using the factorization formula $x^n-a^n=(x-a)(x^{n-1}+a{x}^{n-2}+a^2{x}^{n-3}+\cdots +a^{n-2}x+a^{n-1})$ where $n$ is a positive integer and a is a real number. $$\underset{x\to 2}{lim}\frac{x^5-32}{x-2}$$

Looking ahead to sequences A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence $2,4,6,8,\dots$ is specified by the function $f(n)=2n,$ where $n=1,2,3,\dots ..$ The limit of such a sequence is $\underset{n\to \mathrm{\infty }}{lim}f(n),$ provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. $\text{Missing left or extra right}$ which is defined by $f(n)=\frac{n+1}{n^2},$ for $n=1,2,3,\dots$

Calculate the following limits using the factorization formula $$x^n-a^n=(x-a)(x^{n-1}+a{x}^{n-2}+a^2{x}^{n-3}+\cdots +a^{n-2}x+a^{n-1})$$ where n is a positive integer and a is a real number. $$\underset{x\to 1}{lim}\frac{x^6-1}{x-1}$$

Applying the Intermediate Value Theorem Use the Intermediate Value Theorem to verify that the following equations have three solutions on the given interval. Use a graphing utility to find the approximate roots. $$70x^3-87x^2+32x-3=0;(0,1)$$