Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


llimx→0 (x −x cos x) / sin² 3x

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?

lim x → 0 tan (π/4 cos (sin x¹/³))

Finding Limits

In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.

5 ―x²

lim ------------- = 0

x → ―2 (√g(x))

If f(1)=5, must limx→1 f(x) exist? If it does, then must limx→1 f(x)=5? Can we conclude anything about limx→1 f(x)? Explain.

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?

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Using the Formal Definition

Each of Exercises 31–36 gives a function f(x), a point c, and a positive number ε. Find L = lim x→c f(x). Then find a number δ > 0 such that |f(x)−L| < ε whenever 0 < |x−c| < δ.

f(x) = −3x − 2, c = −1, ε = 0.03

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→∞ √(x² + 1) / (x + 1)