Evaluate the limit of (sqrt(x + 5) - 3)/(x - 4) as x approaches 4.

Step-by-step solution

1. 1

   Verify that it really is 0/0. At $x=4$, $\sqrt{9}-3=0$ and $4-4=0$. Because the radical actually equals the constant here, the conjugate trick will work — unlike cases where the numerator tends to a nonzero value.

2. 2

   Multiply by the conjugate. Multiply numerator and denominator by $\sqrt{x+5}+3$, which is never zero near $x=4$.

3. 3

   Simplify the numerator. $\left(\sqrt{x+5}\right)^2-3^2=(x+5)-9=x-4$, so the quotient is $\frac{x-4}{(x-4)\left(\sqrt{x+5}+3\right)}$.

4. 4

   Cancel and reduce. For $x\neq 4$ this is $\frac{1}{\sqrt{x+5}+3}$, a function that is continuous at $x=4$.

5. 5

   Substitute. $\frac{1}{\sqrt{9}+3}=\frac{1}{3+3}=\frac{1}{6}$.

6. 6

   Numerical check. Evaluating the original expression at $x=4+10^{-8}$ gives $0.1666667$, matching $\frac{1}{6}$.