1. Limits and Continuity

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx → √3 1/x² = 1/3

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→−3 (x² − 9) / (x + 3) = −6

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x² sin (1/x) = 0

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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)

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, P(2,8)

Finding Limits Graphically

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

f. limx→0 f(x) = 0

Finding Limits Graphically

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

j. limx→2− f(x) = 2

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

Finding Limits Graphically

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

e. limx→1+ f(x) = 1

Finding Limits Graphically

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

c. limx→0− f(x) = 0

Finding Limits Graphically

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

e. limx→0 f(x) exists

Finding Limits Graphically

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

d. limx→1− f(x) = 2

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

lim ( 1 / x²/³ + 2 / (x − 1)²/³ ) as

a. x → 0⁺

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

lim ( 1 / x¹/³ − 1 / (x − 1)⁴/³ ) as

a. x → 0⁺

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = 1/(x − 1)