How to Evaluate a Limit by Substitution

Calculus · Limits

How to Evaluate a Limit by Substitution

Problem

\lim_{x \to 2} \left(x^2 + 3x\right)

Answer

10

Direct substitution is the first test for any limit. It works whenever the expression is continuous at the point, which covers every polynomial.

Step-by-step solution

Substitute x = 2 into the expression.
$2^2 + 3(2)$
Why: Polynomials are continuous everywhere, so the limit equals the function value.
Simplify the expression.
$4 + 6$
Why: Evaluate the power first, then the product.
Write the limit.
$4 + 6 = 10$
Why: The expression approaches 10 as x approaches 2.

Common mistakes

Refusing to substitute because the expression looks complicated. Try substitution first every time.
Substituting before simplifying the power and product in the right order.
Reporting the derivative or the slope instead of the limit value.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

$\lim_{x \to 3} \left(x^2 - 1\right)$

Show answer

$8$
$\lim_{x \to 1} \left(2x + 5\right)$

Show answer

$7$
$\lim_{x \to 2} \left(x^3 - 2\right)$

Show answer

$6$
$\lim_{x \to 3} \frac{x + 1}{x - 1}$

Show answer

$2$
$\lim_{x \to 4} \left(5x - 3\right)$

Show answer

$17$