Limits as x → ∞ or x → −∞


The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.


lim x → ∞ √((8x² − 3) / (2x² + x))

Using the Sandwich Theorem

a. It can be shown that the inequalities 1 − x²/ 6 < (x sin x) / (2−2cos x) < 1 hold for all values of x close to zero (except for x = 0). What, if anything, does this tell you about limx→0 (x sin x) / (2 − 2cos x)?

Give reasons for your answer.

[Technology Exercise] b. Graph y = 1 − (x²/6), y=(x sinx)/(2 − 2cos x), and y = 1 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.

At what points are the functions in Exercises 13–30 continuous?

f(x) = { (x³ − 8)/(x² − 4), x ≠ 2, x ≠ −2

3, x = 2

4, x = −2

Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → 3 (−2 / (x − 3)²) = −∞

The sign-preserving property of continuous functions Let f be defined on an interval (a, b) and suppose that f(c) ≠ 0 at some c where f is continuous. Show that there is an interval (c − δ, c + δ) about c where f has the same sign as f(c).

Suppose that f(x) and g(x) are polynomials in x. Can the graph of f(x)/g(x) have an asymptote if g(x) is never zero? Give reasons for your answer.

Never-zero continuous functions Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer.