1. Limits and Continuity

Which one of the following intervals is not symmetric about x=5?

a.(1, 9)

b.(4, 6)

c.(3, 8)

d.(4.5, 5.5)

What does it mean to say that $\underset{n\to \infty }{lim}{a}_n=8$?

Evaluate the limit: $\underset{h\to 0}{lim}\frac{{(2+h)}^2-4}{h}$

Average Rates of Change

In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.

f(x)=x³+1

a. [2, 3]

Suppose the graph of the function $f$ is shown above. What is $\underset{x\to -1}{lim}f\left(f\right(x\left)\right)$?

Finding Limits Graphically

Which of the following statements about the function y = f(x) graphed here are true, and which are false?

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k. limx→3+ f(x) does not exist.

For which values of p does the improper integral ${\int }_1^{\infty }{x}^{3p+1} dx$ converge?

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limt→0 2t / tan t

Use series to evaluate the limit: $\underset{x\to 0}{lim}\frac{\left(\sin \right(3x)-3x+\frac{9}{2}{x}^3)}{x^5}$

Which of the following explains why a function $f\left(x\right)$ is discontinuous at $x=a$?

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limt→0 sin(1 − cos t) / (1 − cos t)

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2−25 / x−5; a=5

Evaluate the following limit. If the limit does not exist, select 'DNE'.
$$\underset{x\to \infty }{lim}arctan\left(e^x\right)$$

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {x^2+1 if x≤−1

3 if x>−1; a=−1

Given the function $f\left(x\right)=x^2$, find a number $\delta$ > $0$ such that if $|x-1|<\delta$, then $|x^2-1|<\frac{1}{2}$.