91Ó°ÊÓ

Write a delta–epsilon proof that proves that f is continuous on its domain. In each case, you will need to assume that δ is less than or equal to 1.

$f\left(x\right)=x^{−1}$

Prove that the sum rule for limits also applies for limits as $x→∞$: If $\underset{x→∞}{\lim }f\left(x\right)=L$and $\underset{x→∞}{\lim }g\left(x\right)=M$, then$\underset{x→∞}{\lim }\left(f\right(x)+g(x\left)\right)=L+M.$

Write a delta–epsilon proof that proves that $f$is continuous on its domain. In each case, you will need to assume that δ is less than or equal to $1$.

$f\left(x\right)=x^2$

Write each of the inequalities in interval notation:

$\left|\left(x^2-1\right)+3\right|<0.5$

Fill in the blanks to complete each of the following theorem statements:

Power functions are continuous everywhere, which means in terms of limits that .

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

$\underset{x→-1^{-}}{\lim }f\left(x\right)=2,\underset{x→-1^{+}}{\lim }f\left(x\right)=2,f(-1)=1.$

Consider the limit expression

$\underset{x→0^{+}}{\lim }\frac{\ln x}{x}$

Calculate the limit.

Determine which of the given forms are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$\frac{0}{0},\frac{0}{∞},\frac{∞}{0},\frac{1}{0},\frac{0}{1},\frac{∞}{1},\frac{1}{∞},\frac{∞}{∞}$

Consider the sequence$\frac{1}{3},\frac{1}{9},\frac{1}{27},\frac{1}{81},…,\frac{1}{3^k},…$

(a) What happens to the terms of this sequence as k gets larger and larger? Express your answer in limit notation.

(b) Find a sufficiently large value of k so that every term past the kth term of this sequence will be less than 0.0001.

Use interval notation to fill in the blanks that follow. Your

answers will involve$\mathrm{δ},\mathrm{Î\mu },N,\mathrm{and}/\mathrm{or}M$.

If $\underset{x→∞}{\lim }f\left(x\right)=2$then for all localid="1649070162044" $\mathrm{Î\mu }>0$there is somelocalid="1649070122210" $N>0$such that if $x∈\_\_\_\_\_\mathrm{then}∈\_\_\_\_\_$.