Evaluate the limit of (1 - x)/(sqrt(5 - x^2) - 2) as x approaches 1.

Step-by-step solution

1. 1

   Check the form. At $x=1$ the numerator is $0$ and the denominator is $\sqrt{4}-2=0$, so this is $\frac{0}{0}$ and needs algebra.

2. 2

   Rationalise the denominator, not the numerator. The radical is downstairs this time, so multiply top and bottom by $\sqrt{5-x^2}+2$. The denominator becomes $\left(5-x^2\right)-4=1-x^2$.

3. 3

   Rewrite the whole quotient. The expression is now $\frac{(1-x)\left(\sqrt{5-x^2}+2\right)}{1-x^2}$.

4. 4

   Factor and cancel. Since $1-x^2=(1-x)(1+x)$, the troublesome $(1-x)$ cancels, leaving $\frac{\sqrt{5-x^2}+2}{1+x}$ for all $x\neq 1$.

5. 5

   Substitute $x=1$. The reduced form is continuous at $1$: $\frac{\sqrt{4}+2}{2}=\frac{4}{2}=2$.

6. 6

   Confirm numerically. Evaluating the original quotient at $x=1+10^{-8}$ returns $2.0000$, matching the algebraic result exactly.