Limit Method

Use the properties of limits to determine continuity.

Determination Principles

• If $\lim_{x \to x_0} f(x) = f(x_0)$, then the function is continuous at $x_0$
• If the limit does not exist, or the limit is not equal to the function value, then the function is discontinuous at that point

Applied Examples

Example 1: determine the continuity of the function $f(x) = \sin x$ at $x = 0$

Solution:

1. $f(0) = \sin 0 = 0$
2. $\lim_{x \to 0} \sin x = 0$
3. $\lim_{x \to 0} f(x) = f(0)$
4. Conclusion: the function is continuous at $x = 0$

Example 2: determine the continuity of the function $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$

Solution:

1. The function is undefined at $x = 1$
2. $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} (x + 1) = 2$
3. The limit exists but the function value is undefined
4. Conclusion: the function is discontinuous at $x = 1$

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Exercises

Exercise 1

Determine the continuity of $f(x) = \begin{cases} \frac{\sin x}{x}, & x \neq 0 \\ 1, & x = 0 \end{cases}$ at $x = 0$.

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Summary

Symbols Used in This Article