1. Limits and Continuity

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.

lim x →π sin (x/2 + sin x)

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

e. arctan x = O(1)

82. For what values of a and b is

lim(x→0)(tan(2x/x³) + a/x² + sin(bx)/x) = 0?

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→3 h(x)

Limits of quotients

Find the limits in Exercises 23–42.

limt→−2 (−2x − 4) / (x³ + 2x²)

Evaluate each limit. 

lim x→2 √4x+10 / 2x−2

Suppose the rental cost for a snowboard is \$25 for the first day (or any part of the first day) plus \$15 for each additional day (or any part of a day).

Evaluate lim t→2.9 f(t).

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

a. lim x→−2^+ f(x)

Determine the following limits. 

lim x→∞ (3x¹² − 9x⁷)

Additional Graphing Exercises

[Technology Exercise] Graph the curves in Exercises 109–112. Explain the relationship between the curve’s formula and what you see.

y = −1 / √(4 − x²)

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

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

Calculate the following limits using the factorization formula x^n−a^n=(x−a)(x^n−1+ax^n−2+a^2x^n−3+⋯+a^n−2x+a^n−1), where n is a positive integer and a is a real number.

lim x→1 x^6 − 1 / x − 1

Determine the following limits.

b. lim t→−2^− t^3 − 5t^2 + 6t / t^4 − 4t^2

Use analytic methods to find the value of lim x→π/4 cos 2x / cos x − sin x.

Evaluate each limit. 

lim x→e^2 ln^2x−5 ln x+6 lnx−2