Evaluate a Limit by Rationalizing

Evaluate a limit by rationalizing

When a limit contains a square root, direct substitution may give an indeterminate form such as 0/0. Rationalizing helps remove the radical by using a conjugate.

1) Try direct substitution first

Plug the target value into the expression. If the result is a normal number, that is the limit. If you get 0/0, rationalizing is a good next step.

2) Multiply by the conjugate

If the expression has a difference like \sqrt{a(x)} - \sqrt{b(x)}, multiply numerator and denominator by the conjugate \sqrt{a(x)} + \sqrt{b(x)}. This uses the identity (u-v)(u+v)=u^2-v^2. That removes the square roots in the numerator.

3) Simplify carefully

After multiplying, cancel common factors if possible. Often a factor such as (x-c) appears and can be simplified away. Then substitute the limit value again.

4) Check the result

Make sure the final expression is defined near the limit point and that the simplified form gives a finite value. If possible, verify by substituting nearby values to see whether the expression approaches your answer.

Common tip

Only rationalize the part that causes the indeterminate form. Keep track of signs when the radical is inside a difference.