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→∞ (x − 3) / √(4x² + 25)

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0− h / sin 3h

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)

lim x → ±∞ f(x) = 0, lim x → 2⁻ f(x) = ∞, and lim x → 2⁺ f(x) = ∞

Horizontal and Vertical Asymptotes

Determine the domain and range of y = (√16―x²) / (x―2).

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0 sin(sin h) / sin h

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 sin θ cot 2θ

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.