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?

e. limx→1+ f(x) = 1

For which values of $x$ does the series $\sum _{n=1}^{\infty }\frac{x^n}{n}$ converge?

Using the Sandwich Theorem

a. Suppose that the inequalities 1/2 − x² / 24 < (1 − cos x)/ x² < 1/2 hold for values of x close to zero, except for x = 0 itself. (They do, as you will see in Section 9.9.) What, if anything, does this tell you about limx→0 (1 −cos x)/ x²?

Give reasons for your answer.

[Technology Exercise] b. Graph the equations y=(1/2) − (x²/24), y = (1 - cos x) / x², and y = 1/2 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.

The accompanying graph shows the total amount of gasoline A in the gas tank of an automobile after it has been driven for t days.

c. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs.

Determine the following limits at infinity.

lim x→ ∞ x^−6

Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → −5 (1 / (x + 5)²) = ∞

Finding Limits

In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)

g(x) = 1/(2 + (1/x))

Complete the following steps for the given functions. 

a. Find the slant asymptote of $f$.

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

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

f(x)=x^2−3x+2 / x^10−x^9

Explain the meaning of lim x→−∞ f(x)=10.

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

lim x→∞ cot^−1

Complete the following steps for the given functions. 

a. Find the slant asymptote of $f$.

$f\left(x\right)=\frac{3x^2-2x+5}{3x+4}$

Suppose x lies in the interval (1, 3) with x≠2. Find the smallest positive value of δ such that the inequality 0<|x−2|<δ is true.

Suppose that a function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of limx→0 f(x)? Give reasons for your answer.

The function $s(t)$ represents the position of an object at time t moving along a line. Suppose $s(2)=136$ and $s(3)=156$ . Find the average velocity of the object over the interval of time $[2, 3]$ .