1. Limits and Continuity

Finding Limits Graphically

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

f. limx→0 f(x) = 0

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

a. f(1)

Complete the following steps for the given functions. 

b. Find the vertical asymptotes of $f$ (if any).

$f\left(x\right)=\frac{x^2-2x+5}{3x-2}$

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (tan 2x) / x

a. Use a graphing utility to estimate lim x→0 tan 2x / sin x, lim x→0 tan 3x / sin x, and lim x→0 tan 4x / sin x.

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→⁻∞ (³√x − ⁵√x) / (³√x + ⁵√x)

A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.

a. lim x→−2^+ f(x)

Use the graph of g(x) in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

${\lim }_{x\to 4}g\left(x\right)$

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let f(x) = (x² - 9) / (x + 3)

b. Support your conclusions in part (a) by graphing f near c = -3 and using Zoom and Trace to estimate y-values on the graph as x → −3.

Use a graph of f to estimate $\underset{x\to a}{\lim }f\left(x\right)$ or to show that the limit does not exist. Evaluate f(x) near $x=a$ to support your conjecture.

$f\left(x\right)=\frac{x-2}{\ln \mid x-2\mid }$; $a=2$

Given the graph of f in the following figures, find the slope of the secant line that passes through (0,0) and (h,f(h))in terms of h, for h>0 and h<0.

f(x)=x^1/3 <IMAGE>

If f(1)=5, must limx→1 f(x) exist? If it does, then must limx→1 f(x)=5? Can we conclude anything about limx→1 f(x)? Explain.

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let h(x)=(x² − 2x − 3)/(x² − 4x + 3)

b. Support your conclusions in part (a) by graphing h near c = 3 and using Zoom and Trace to estimate y-values on the graph as x→3.

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

llimx→0 (x −x cos x) / sin² 3x

Evaluate lim x→∞ f(x) and lim x→−∞ f(x) sing the figure. <IMAGE>