1. Limits and Continuity

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

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0− h / sin 3h

Finding Limits Graphically

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

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g. limx→0+ f(x) = limx→0− f(x)

Finding Limits Graphically

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

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k. limx→3+ f(x) does not exist.

Finding Limits Graphically

Let f(x) = {3 - x , x < 2

2, x = 2

x/2, x > 2

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a. Find limx→2+ f(x), limx→2− f(x), and f(2).

Finding Limits Graphically

Let f(x) = {3 - x , x < 2

2, x = 2

x/2, x > 2

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c. Find limx→−1− f(x) and limx→−1+ f(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?