Limits by Rationalization - Calculus Step by Step

Evaluate a limit involving a square root by multiplying by the conjugate to clear the radical.

limit-rationalize When the numerator contains a square root, multiply the numerator and denominator by the conjugate. The product of conjugates eliminates the radical, allowing cancellation and substitution.

Example

Use the conjugate to simplify a radical limit before substituting.

highlighted = computed this step

Step 1 — Set up

Set up the radical limit at x = 0.

\lim_{x\to 0 } \frac{\sqrt{x+4}-2}{x}

Step 2 — Check the form

Direct substitution gives 0 over 0.

\hlmath{\frac{0}{0}}

Step 3 — Multiply by the conjugate

Multiply by the conjugate root x plus 4 plus 2.

\frac{\sqrt{x+ \hl{4} }+ \hl{2} }{\sqrt{x+ 4 }+ 2 }

Step 4 — Simplify

Simplify the rationalized expression.

\hlmath{\frac{1}{\sqrt{x+4}+2}}

Step 5 — Substitute

Substitute 0 to get 1 over 4.

\lim= \hlmath{\frac{1}{4}}