Question

1.7 selecting procedures for d
calculus
evaluate each imit.

1. $\lim_{x \to 1} \frac{x - 1}{\sqrt{x} - 1}$

Explanation:

Step1: Rationalize the denominator

Multiply the numerator and denominator by $\sqrt{x}+1$.

$$\lim_{x ightarrow1}\frac{(x - 1)(\sqrt{x}+1)}{(\sqrt{x}-1)(\sqrt{x}+1)}$$

Using the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, the denominator becomes $x - 1$.
So the limit is $\lim_{x
ightarrow1}\frac{(x - 1)(\sqrt{x}+1)}{x - 1}$.

Step2: Simplify the expression

Cancel out the common factor $(x - 1)$ (for $x
eq1$, as we are taking the limit as $x
ightarrow1$).
We get $\lim_{x
ightarrow1}(\sqrt{x}+1)$.

Step3: Evaluate the limit

Substitute $x = 1$ into $\sqrt{x}+1$.
When $x = 1$, $\sqrt{1}+1=1 + 1=2$.

Answer:

$2$