1. Limits and Continuity

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

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

Calculate $f\(\left\)(x\(\right\))$ for each value of $x$ in the following table.

Slope of a Curve at a Point

In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.

y=x³−3x²+4, P(2,0)

Suppose |f(x) − 5|<0.1 whenever 0<x<5. Find all values of δ>0 such that |f(x) − 5|<0.1 whenever 0<|x−2|<δ.

[Technology Exercise] Grinding engine cylinders Before contracting to grind engine cylinders to a cross-sectional area of 9in², you need to know how much deviation from the ideal cylinder diameter of c = 3.385in. you can allow and still have the area come within 0.01in² of the required 9in². To find out, you let A=π(x/2)² and look for the largest interval in which you must hold x to make |A − 9| ≤ 0.01. What interval do you find?

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 → ⁻∞ ((1 − x³) / (x² + 7x))⁵

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→9 √(x − 5) = 2

Using the graph, find the specified limit or state that the limit does not exist (DNE).

${\lim }_{x\to -2^{-}}f\left(x\right)$, ${\lim }_{x\to -2^{+}}f\left(x\right)$, ${\lim }_{x\to -2^{}}f\left(x\right)$

Finding Limits Graphically

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

b. limx→2 f(x) does not exist

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

lim x → 3 (−2 / (x − 3)²) = −∞

A rock is dropped off the edge of a cliff, and its distance s (in feet) from the top of the cliff after t seconds is s(t)=16t^2. Assume the distance from the top of the cliff to the ground is 96 ft.

a. When will the rock strike the ground? 

Average Rates of Change

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

g(x)=x²−2x

a. [1, 3]

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

f(−1)

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

lim x→1 f(x)

Estimating Limits

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

Let g(θ) = (sinθ) / θ.

b. Support your conclusion in part (a) by graphing g near θ₀ = 0.