Limits

Limits by Substitution

Find a limit by direct substitution when the function is continuous at the approach point.

Example

Substitute the approach value when the function is continuous there.

highlighted = computed this step

Step 1 — Set up

Set up the direct-substitution limit at x = 2.

lim⁡x→2x2+3x−1\lim_{x\to 2 } x^{2}+3x-1

Step 2 — Substitute

Substitute 2 for x in each term.

22+3⋅2−1\hl{2} ^{ 2 }+ 3 \cdot \hl{2} - 1

Step 3 — Compute

Compute the substituted value: 9.

lim⁡=9\lim= \hl{9}

Step 4 — Result

State the limit value as 9.

lim⁡x→2x2+3x−1=9\lim_{x\to 2 } x^{2}+3x-1 = \hl{9}
limit-by-substitution If f is continuous at x = a, then lim_{x→a} f(x) = f(a). Substitute x = a directly into the expression to find the limit.