The Limit — Question 1

Question 1 - Solution

We are asked to evaluate: $$\lim_{x \to 1} \frac{x^2 - 1}{\sqrt{x} - 1}$$

This limit gives an indeterminate form $\frac{0}{0}$, so we must simplify.

Start by factoring the numerator: $$x^2 - 1 = (x - 1)(x + 1)$$

Now rewrite the expression: $$\frac{(x - 1)(x + 1)}{\sqrt{x} - 1}$$

To simplify, multiply numerator and denominator by the **conjugate** of the denominator: $$\frac{(x - 1)(x + 1)}{\sqrt{x} - 1} \cdot \frac{\sqrt{x} + 1}{\sqrt{x} + 1}$$

This gives: $$\frac{(x - 1)(x + 1)(\sqrt{x} + 1)}{x - 1}$$

Now cancel $x - 1$ (which is nonzero near $x = 1$): $$(x + 1)(\sqrt{x} + 1)$$

Now substitute $x = 1$: $$(1 + 1)(\sqrt{1} + 1) = 2 \cdot 2 = \boxed{4}$$