1 / 5
Limits and Continuity
Limits and Continuity
Calculus begins with the idea of a limit: what value a function approaches as the input approaches a target.
Definition
We write $$\lim_{x \to a} f(x) = L$$ when the values of $f(x)$ get arbitrarily close to $L$ as $x$ gets close to $a$.
Limit Laws
- The limit of a sum is the sum of the limits.
- The limit of a product is the product of the limits.
- For polynomials, substitute directly.
- For rational functions, direct substitution works if the denominator is not zero.
Example 1
Evaluate $$\lim_{x \to 3}(2x+5).$$
Substitute directly: $2(3)+5=11$.
Example 2
Evaluate $$\lim_{x \to 2} \frac{x^2-4}{x-2}.$$
Factor: $$\frac{(x-2)(x+2)}{x-2}=x+2$$ for $x \ne 2$. The limit is $4$.
Continuity
A function is continuous at $x=a$ if $f(a)$ exists, the limit exists, and the two are equal.
Example 3
Is $f(x)=x^2-1$ continuous at $x=4$?
Yes. Polynomials are continuous everywhere.
Practice Problems
1. $$\lim_{x \to 1}(3x-7)$$
2. $$\lim_{x \to -2}(x^2+4x)$$
3. $$\lim_{x \to 5} \frac{x^2-25}{x-5}$$
4. Decide whether $f(x)=\frac{1}{x-3}$ is continuous at $x=3$.
5. State a point where $f(x)=|x|$ is continuous.
6. What does a removable discontinuity look like algebraically?
Show Answer Key
1. $-4$
2. $-4$
3. $10$
4. No, denominator is zero there.
5. Everywhere; for example at $x=0$.
6. A common factor cancels, leaving a hole in the graph.