Limits by Rationalization: Step-by-Step Guide
When a limit gives 0/0 and the expression contains a square root, multiplying by the conjugate clears the radical from one part of the fraction and exposes a common factor that can be canceled. This is the standard technique for limits like (sqrt(x+a) - b)/(x - c) at points where direct substitution fails.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Confirm the indeterminate form 0/0. Substitute the target value to confirm the form is 0/0. If a radical sits in numerator or denominator, rationalization is the right tool.\[\lim_{x \to a} \frac{\sqrt{f(x)} - b}{g(x)} \to \frac{0}{0}\]
- Multiply by the conjugate over itself. If the radical is sqrt(u) - v, multiply numerator and denominator by sqrt(u) + v. If sqrt(u) + v, multiply by sqrt(u) - v. This uses (a-b)(a+b) = a^2 - b^2 to remove the square root.\[\frac{\sqrt{u} - v}{w} \cdot \frac{\sqrt{u} + v}{\sqrt{u} + v} = \frac{u - v^2}{w(\sqrt{u} + v)}\]
- Simplify and cancel. After multiplying, the numerator becomes a polynomial. Factor and cancel the (x - a) common factor with the denominator.\[\frac{u - v^2}{w(\sqrt{u} + v)} \to \text{cancel } (x - a)\]
- Substitute the target value. Plug x = a into the simplified expression to obtain the limit.\[\lim_{x \to a} \text{(simplified)} = L\]
Worked example
Evaluate lim x->0 (sqrt(x + 4) - 2)/x.
Step 1: Direct substitution gives (sqrt(4) - 2)/0 = 0/0, indeterminate.
Step 2: Multiply numerator and denominator by the conjugate sqrt(x + 4) + 2: ((sqrt(x+4) - 2)(sqrt(x+4) + 2))/(x(sqrt(x+4) + 2)).
Step 3: The numerator simplifies via (a-b)(a+b) to (x + 4) - 4 = x. The expression becomes x/(x(sqrt(x+4) + 2)).
Step 4: Cancel x: 1/(sqrt(x+4) + 2).
Step 5: Substitute x = 0: 1/(sqrt(4) + 2) = 1/(2 + 2) = 1/4.
Verification: At x = 0.01, (sqrt(4.01) - 2)/0.01 approximately (2.0025 - 2)/0.01 = 0.2497, very close to 0.25 = 1/4.
Tips and common mistakes
- Use rationalization whenever a square root appears and substitution gives 0/0.
- The conjugate of a + b is a - b and vice versa; always multiply numerator and denominator by the same conjugate.
- After multiplying, expand only the part where the radical lives; leave the other side factored to spot cancellation.
- If both numerator and denominator have radicals, you may need to rationalize twice.
- (a - b)(a + b) = a^2 - b^2 is the key identity — it eliminates square roots cleanly.
- For cube roots, use the identity a^3 - b^3 = (a - b)(a^2 + ab + b^2) instead of a simple conjugate.
- Always check that the canceled factor really equals x - a before crossing it out.
- Plug in a value close to the target (like 0.01) to numerically confirm the final answer.