1. Limits and Continuity

Centering Intervals About a Point

In Exercises 1–6, sketch the interval (a,b), on the x-axis with the point c inside. Then find a value of δ>0 such that a < x < b whenever 0 < |x−c| < δ.

a=4/9, b=4/7, c=1/2

Use the graph of $f\(\left\)(x\(\right\))$to estimate the value of the limit or state that it does not exist (DNE).

$\(\lim\)_{x\(\rarr\)0}f\(\left\)(x\(\right\))$

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

$\underset{x\to 0}{\lim }(\frac{1}{x^2}+1)=\infty$

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→3 (3x − 7) = 2

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

lim ( 1 / x¹/³ − 1 / (x − 1)⁴/³ ) as

a. x → 0⁺

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (1 − cos 3x) / 2x

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

d. limx→c f(x) exists at every point c in (-1,1).

Finding Limits of Differences When x → ±∞

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

lim x → ∞ (√(9x² − x) − 3x)

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

b. lim x→0⁻ 2 / (3x¹/³)

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^+ tan x

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

lim x→3^− f(x)

Determine ${\lim }_{x\to \infty }f\left(x\right)$ and ${\lim }_{x\to -\infty }f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{4x}{20x+1}$

Determine $\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\))$ and $\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\))$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{3x^3-7}{x^4+5x^2}$

Tangent lines with zero slope

a. Graph the function f(x)=x^2−4x+3.

Average Rates of Change

In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.

h(t)=cot t

a. [π/4,3π/4]