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Evaluate a limit by rationalizing the denominator

Consider the function f(x)=x2−1x−1f(x) = \dfrac{x^2 - 1}{\sqrt{x} - 1}.

Evaluate the limit lim⁡x→1x2−1x−1.\lim_{x \to 1} \frac{x^2 - 1}{\sqrt{x} - 1}. Note that direct substitution gives the indeterminate form 0/00/0, so you will need to remove the algebraic obstruction before evaluating.

Hint

Multiply the numerator and denominator by the conjugate x+1\sqrt{x} + 1 to eliminate the square root in the denominator, then simplify and cancel common factors before substituting x=1x = 1.

Step-by-step solution

Multiply numerator and denominator by the conjugate x+1\sqrt{x}+1: x2−1x−1⋅x+1x+1=(x2−1)(x+1)x−1.\frac{x^2-1}{\sqrt{x}-1}\cdot\frac{\sqrt{x}+1}{\sqrt{x}+1} = \frac{(x^2-1)(\sqrt{x}+1)}{x-1}. This clears the radical from the denominator, leaving a factor of (x−1)(x-1) there.

Factor the numerator using x2−1=(x−1)(x+1)x^2-1=(x-1)(x+1), giving (x−1)(x+1)(x+1)x−1.\frac{(x-1)(x+1)(\sqrt{x}+1)}{x-1}. Since x→1x\to 1 but x≠1x\ne 1, cancel the common factor (x−1)(x-1) to get (x+1)(x+1)(x+1)(\sqrt{x}+1).

Now substitute x=1x=1 directly into the simplified expression: (1+1)(1+1)=2⋅2=4(1+1)(\sqrt{1}+1) = 2\cdot 2 = 4. So the limit equals 44.

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