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

Finding Deltas Graphically

In Exercises 7–14, use the graphs to find a δ>0 such that |f(x)−L| <ε whenever 0< |x−c| <δ.

Finding Deltas Algebraically

Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.

f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

Finding Deltas Algebraically

Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.

f(x) = mx, m > 0, L = 2m, c = 2, ε = 0.03

Using the Formal Definition

Each of Exercises 31–36 gives a function f(x), a point c, and a positive number ε. Find L = lim x→c f(x). Then find a number δ > 0 such that |f(x)−L| < ε whenever 0 < |x−c| < δ.

f(x) = −3x − 2, c = −1, ε = 0.03

[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?

Finding Limits Graphically

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

a. limx→−1+ f(x) = 1

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=7−x², P(2,3)

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = x², x = 1

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limt→0 2t / tan t

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (x² − x + sin x) / 2x

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limt→0 sin(1 − cos t) / (1 − cos t)

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 sin θ / sin 2θ

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 θcos θ

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (1 − cos 3x) / 2x