91Ó°ÊÓ

what we mean when we say that a limit exists, or that a limit does not exist.

Calculating limits: Find each limit by hand.

$\underset{x→1}{\lim }\frac{2x^3-x^2-2x+1}{x^2-2x+1}$.

Exercises 3–6, $\underset{x→c}{\lim }f\left(x\right)=L$In and $\underset{x→c}{\lim }g\left(x\right)=M$for some

real numbers L and M. What, if anything, can you say about $\underset{x→c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$in each case?

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive

$\underset{x→∞}{\lim }{x}^k=?$

Explain why we can’t calculate every limit $\underset{x→c}{\lim }f\left(x\right)$ just by evaluating f(x) at $x=c$. Support your argument with the graph of a function f for which $\underset{x→c}{\lim }f\left(x\right)=f\left(c\right)$.

In our proof that constant functions are continuous, we used the fact that given any $∈$> 0, a choice of any δ > 0 will work in the formal definition of limit. Use a graph to explain why this makes intuitive sense. (This exercise depends on Section 1.3.)

Find punctured intervals on which the function $f\left(x\right)=\frac{1}{x^2-x}$is defined, centered around

$\left(\mathrm{a}\right)x=1.5\left(\mathrm{b}\right)x=0.25\left(\mathrm{c}\right)x=1$

The Binomial Theorem says that an expression of the form ${\left(a+b\right)}^n$can be expanded to localid="1649785698964" $\left(\begin{array}{c}n\\ 0\end{array}\right)a^n{b}^0+\left(\begin{array}{c}n\\ 1\end{array}\right)a^{n-1}{b}^1+\left(\begin{array}{c}n\\ 2\end{array}\right)a^{n-1}{b}^2+⋯+\left(\begin{array}{c}n\\ n\end{array}\right)x^0{y}^n,$where for any localid="1649783317676" $0≤k≤n,$the symbol $\left({{}_k}^n\right)$is equal to $\frac{n!}{k!\left(n-k\right)!}.$Here $n!$is n factorial, the product of the integers from $1$to n. By convention we set $0!=1.$Apply this expansion to the expressionlocalid="1649783538047" ${\left(1+\frac{1}{n}\right)}^n.$

Use the Extreme Value Theorem to show that each function f has both a maximum and a minimum value on [a, b]. Then use a graphing utility to approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere.

$f\left(x\right)=x^4−3x^2−2,[a,b]=[0,2]$

For each limit $\underset{x→c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\mathrm{δ}$such that if $x∈(c-\mathrm{δ},c)∪(c,c+\mathrm{δ})\mathrm{then}\mathrm{f}\left(\mathrm{x}\right)∈(\mathrm{L}-\mathrm{Î\mu },\mathrm{L}+\mathrm{Î\mu })\mathit{}$.

$\underset{\mathrm{x}→0}{\lim }\frac{1-\cos \left(\mathrm{x}\right)}{\mathrm{x}}=0,\mathrm{Î\mu }=\frac{1}{4}$