1. Limits and Continuity

Theory and Examples

a. If limx→0 f(x) / x² = 1, find limx→0 f(x).

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (3x⁴ + 3x³ − 36x²) / (x⁴ − 25x² + 144)

Suppose p and q are polynomials. If lim x→0 p(x) / q(x)=10 and q(0)=2, find p(0).

Determine the following limits.

lim x→a (3x + 1)^2 − (3a + 1)^2 / x − a, where a is constant

Determine the following limits. 

lim x→∞ 6x²/(4x^2+√(16x⁴ + x²))

Determine the following limits.

c. lim x→4 x − 5 / (x − 4)^2

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x²/2 − 1/x) as

b. x→0⁻

10. True, or false? As x→∞,

c. 1/x - 1/x² = o(1/x)

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

57. lim (x → 0⁺) x^(-1/ln x)

Determine the following limits. 

lim θ→∞ cos θ / θ²

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 3x + 2) / (x³ − 4x) as

b. x→−2⁺

82. Use the definitions of the hyperbolic functions to find each of the following limits.

h. lim(x→0^-) coth x

Oblique Asymptotes

Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.

y = x² / (x − 1)

109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.

b. f(x)=x, g(x)=x + 1/x

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.

The limit lim x→a f(x) / g(x) does not exist if g(a)=0.