1. Limits and Continuity

Use the graph of $f$ in the figure to find the following values or state that they do not exist. <IMAGE>

$f\(\left\)(0\(\right\))$

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→a (mx+b)=ma+b, for any constants a, b, and m

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→−1^− f(x)

Estimate the following limits using graphs or tables.

lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all vertical asymptotes.

x² + x ― 6

c. y = ------------------

x² + 2x ― 8

Use the precise definition of infinite limits to prove the following limits.

$\underset{x\to -1}{\lim }\frac{1}{{(x+1)}^4}=\infty$

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = x², x = -2

[Technology Exercise] Let f(t) = 1/t for t≠0.

a. Find the average rate of change of f with respect to t over the intervals (i) from t=2 to t=3, and (ii) from t=2 to t=T.

b. Make a table of values of the average rate of change of f with respect to t over the interval [2,T], for some values of T approaching 2, say T = 2.1, 2.01, 2.001, 2.0001, 2.00001, and 2.000001.

c. What does your table indicate is the rate of change of f with respect to t at t=2?

Use the graph of $h$ in the figure to find the following values or state that they do not exist. <IMAGE>

$h\(\left\)(2\(\right\))$

Find the limit using the graph of $f\(\left\)(x\(\right\))$shown.

$\(\lim\)_{x\(\rarr\)1}f\(\left\)(x\(\right\))$

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→1 (8x+5)=13

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time. 

c. s(t)=40 sin 2t at t=0

The accompanying graph shows the total distance s traveled by a bicyclist after t hours.

b. Estimate the bicyclist’s instantaneous speed at the times t=1/2, t=2, and t=3.

Use the graph of the greatest integer function y = ⌊x⌋, Figure 1.10 in Section 1.1, to help you find the limits in Exercises 21 and 22.

<IMAGE>

b. limt→4−(t−⌊t⌋)

Finding Limits

For the function f whose graph is given, determine the following limits. Write ∞ or −∞ where appropriate.

h. lim x → ∞ f(x)