1. Limits and Continuity

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0 sin(sin h) / sin h

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 sin θ cot 2θ

The position of an object moving vertically along a line is given by the function $s\(\left\)(t\(\right\))=-4.9t^2+30t+20$. Find the average velocity of the object over the following intervals.

$\(\left\[\lbrack\)0,3\(\right\]\rbrack\)$

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 √(4 − x) = 2

Estimating Limits

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

Let F(x)=(x² + 3x + 2)/(2−|x|)

b. Support your conclusion in part (a) by graphing F near c = -2 and using Zoom and Trace to estimate y-values on the graph as x→−2.

Determine the following limits.

Assume the function g satisfies the inequality 1≤g(x) ≤sin^2 x + 1, for all values of x near 0. Find lim x→0 g(x).

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→1 1/x = 1

63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.

limₕ→₀ (1 + 3h)^{2/h}

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

f(1)

The position of an object moving vertically along a line is given by the function $s\(\left\)(t\(\right\))=-16t^2+128t$. Find the average velocity of the object over the following intervals.

$\(\left\[\lbrack\)1,4\(\right\]\rbrack\)$

Suppose ${\(\displaystyle\]\lim\)_{x\(\to\) a}}\(\text{ }\)f\(\left\)(x\(\right\))=L$. Prove that $\underset{x\to a}{\lim }\text{ (}c\left(f\left(x\right)\right)=cL$, where $c$ is a constant.

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→2 (x^2+3x)=10

Determine whether the following statements are true and give an explanation or counterexample.

c. The graph of a function can have any number of vertical asymptotes but at most two horizontal asymptotes.

Finding Limits of Differences When x → ±∞

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

lim x → −∞ (√(x² + 3) + x)

Which of the following statements about the series $\sum \underset{}{n=1}\infty \frac{1}{2^n}$ is true?