Limits of Composite Functions

The limit of a composite function is an important topic in limit operations. It describes how to compute the limit of a function formed by composing two functions.

Theorem

Conditions of the Theorem

1. The Limit of the Inner Function Exists

$\lim_{x \to x_0} f(x) = A$ must exist

2. The Outer Function Has a Limit at the Inner Limit

$\lim_{u \to A} g(u) = B$ must exist

3. The Inner Function Does Not Equal Its Limit

$f(x) \neq A$ (when $x \neq x_0$)

This condition guarantees that during the limiting process, $u = f(x)$ never reaches $A$, so that $\lim_{u \to A} g(u) = B$ can be applied.

Worked Examples

Example 1

Compute $\lim_{x \to 0} \sin(x^2)$

Solution:

1. Let $u = x^2$; then as $x \to 0$, $u \to 0$

2. $\lim_{x \to 0} \sin(x^2) = \lim_{u \to 0} \sin u = 0$

Example 2

Compute $\lim_{x \to 1} e^{x-1}$

Solution:

1. Let $u = x - 1$; then as $x \to 1$, $u \to 0$

2. $\lim_{x \to 1} e^{x-1} = \lim_{u \to 0} e^u = e^0 = 1$

Example 3

Compute $\lim_{x \to 0} \ln(1 + x^2)$

Solution:

1. Let $u = x^2$; then as $x \to 0$, $u \to 0$

2. $\lim_{x \to 0} \ln(1 + x^2) = \lim_{u \to 0} \ln(1 + u) = \ln 1 = 0$

Composition of Continuous Functions

If $g(u)$ is continuous at $u = A$, i.e., $\lim_{u \to A} g(u) = g(A)$, then:

This conclusion shows that for continuous functions, the limit symbol and the function symbol can be interchanged.

Examples

• $\lim_{x \to 0} e^{x^2} = e^{\lim_{x \to 0} x^2} = e^0 = 1$
• $\lim_{x \to 1} \sin(x - 1) = \sin(\lim_{x \to 1} (x - 1)) = \sin 0 = 0$
• $\lim_{x \to 0} \ln(1 + x^2) = \ln(\lim_{x \to 0} (1 + x^2)) = \ln 1 = 0$

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Practice Problems

Exercise 1

Compute the limit $\lim_{x \to 0} \cos(x^2)$.

Exercise 2

Compute the limit $\lim_{x \to 2} e^{x^2 - 4}$.

Exercise 3

Compute the limit $\lim_{x \to 0} \ln(1 + \sin x)$.

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Summary

Symbols Used in This Article

中英对照