1. Limits and Continuity

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = −3/(x − 3)

Let $g\(\left\)(x\(\right\))=\(\frac{x^3-4x}{8\left|x-2\right|}\)$. <IMAGE>

Make a conjecture about the values of ${\(\displaystyle\]\lim\)_{x\(\to\)2^{-}}g\(\left\)(x\(\right\))}$, ${\(\displaystyle\]\lim\)_{x\(\to\)2^{+}}g\(\left\)(x\(\right\))}$, and ${\(\displaystyle\]\lim\)_{x\(\to\)2}g\(\left\)(x\(\right\))}$ or state that they do not exist.

Estimate the following limits using graphs or tables.

$\underset{h\to 0}{\lim }\frac{\ln (1+h)}{h}$

Find the limit by creating a table of values.

$\(\lim\)_{x\(\rarr\)1}\(\frac{x^2-4}{x-2}\)$

The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon.

a. Estimate the slopes of the secant lines PQ₁, PQ₂, PQ₃, and PQ₄, arranging them in a table like the one in Figure 2.6.

b. About how fast was the object going when it hit the surface?

Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → ∞ (√(x² + x) − √(x² − x))

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

lim x → 0 (1 / |x|) = ∞

Finding Limits Graphically

Graph the functions in Exercises 9 and 10. Then answer these questions.

f(x) = {x,−1 ≤ x < 0, or 0 < x ≤ 1

1, x = 0

0, x < −1 or x > 1

d. At what points does the right-hand limit exist but not the left-hand limit?

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = 1/(x − 1)

Using the Formal Definitions

Use the formal definitions of limits as x → ±∞ to establish the limits in Exercises 91 and 92.

If f has the constant value f(x) = k, then lim x → ∞ f(x) = k.

Find the limit by creating a table of values.

$\(\lim\)_{x\(\rarr\)2}3x^2+5x+1$

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers and assume lim x→a f(x) =L

d. If |x−a|<δ, then a−δ<x<a+δ.

Given the function $f\(\left\)(x\(\right\))=-16x^2+64x$, complete the following. <IMAGE>

Find the slopes of the secant lines that pass though the points $\(\left\)(x,f\(\left\)(x\(\right\))\(\right\))$ and $\(\left\)(2,f\(\left\)(2\(\right\))\(\right\))$, for $x=1.5,1.9,1.99,1.999,$ and $1.9999$ (see figure).

Determine the following limits at infinity.

lim x→−∞ x^−11

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

${\(\displaystyle\]\lim\)_{x\(\to\)0}g\(\left\)(x\(\right\))}$