Evaluate limits algebraically

Algebraic evaluation of limits involves determining the value a function approaches as the input nears a specified finite number, using direct substitution for continuous polynomial, rational, and radical expressions and algebraic simplification when substitution produces an indeterminate form such as $0/0$ . Factoring and canceling common factors or rationalizing can reveal a removable discontinuity, emphasizing that a limit describes nearby behavior and need not equal the function’s value at the point; this understanding supports continuity and the definition of the derivative. Formal $\varepsilon$ - $\delta$ proofs, multivariable limits, and more advanced generalized cases are not included.

Detailed Explanation: Evaluate limits algebraically

To evaluate a limit algebraically, first try direct substitution: replace the variable with the number it approaches.

If substitution gives a real number, that is the limit.
If substitution gives an indeterminate form such as $\frac{0}{0}$ , simplify the expression algebraically before substituting again.

Example

Evaluate

\lim_{x\to 3}\frac{x^2-9}{x-3}.

Step 1: Try direct substitution.

Substitute $x=3$ :

\frac{3^2-9}{3-3} = \frac{9-9}{0} = \frac{0}{0}.

Since $\frac{0}{0}$ is indeterminate, we need to simplify.

Step 2: Factor the numerator.

The numerator is a difference of squares:

x^2-9=(x-3)(x+3).

So the expression becomes

\frac{(x-3)(x+3)}{x-3}.

Step 3: Cancel the common factor.

For values of $x$ near $3$ but not equal to $3$ , we can cancel $x-3$ :

\frac{(x-3)(x+3)}{x-3}=x+3.

Therefore,

\lim_{x\to 3}\frac{x^2-9}{x-3} = \lim_{x\to 3}(x+3).

Step 4: Substitute again.

3+3=6.

Thus,

\boxed{\lim_{x\to 3}\frac{x^2-9}{x-3}=6}.

The original function is undefined at $x=3$ , but the limit is $6$ because the nearby values approach $6$ . A limit describes what happens near the input, not necessarily the function’s value exactly at that input.

Learn by doing: Evaluate limits algebraically

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Limits - Continuous Function (Substitution) to Limit

Earned ?