1. Limits and Continuity

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→0 √(7 + sec²x)

Determine the following limits.

$\underset{\theta \to {\frac{\pi }{2}}^{+}}{\lim }\frac{1}{3}\tan \theta$

Determine the following limits.

lim h→0 √5x + 5h − √5x / h, where x>0 Is constant

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim h→0 √16 + h − 4 / h

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

75. lim (x → ∞) e^(x²) / (x e^x)

111. True, or false? Give reasons for your answers.

c. x = o(x + ln(x))

Suppose g(x)=f(1−x) for all x, lim x→1^+ f(x)=4, and lim x→1^− f(x)=6. Find lim x→0^+ g(x) and lim x→0^− g(x).

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→0 (2sin x − 1)

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

f(x) = (x² − 9)/(x(x−3))

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

f(x) = (√(16x⁴ + 64x²) + x²) / (2x² − 4) 

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→2 f (x)

Determine the following limits.

lim x→∞ (x⁴ − 1) / (x⁵ + 2)

Determine the following limits.

$\underset{x\to 0}{\lim }\frac{x^3-5x^2}{x^2}$

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).

f(x) = (2x³ + 10x² + 12x) / (x³ + 2x²)

{Use of Tech} Computing limits with angles in degrees Suppose your graphing calculator has two functions, one called sin x, which calculates the sine of x when x is in radians, and the other called s(x), which calculates the sine of x when x is in degrees.

b. Evaluate lim x→0 s(x) / x. Verify your answer by estimating the limit on your calculator.