Find the value of the limit $\lim _ { x \rightarrow 1 } \frac { x - 1 } { \sqrt { x } - 1 }$
A)
$\sqrt { 2 }$
B)1
C)-1
D) $- \sqrt { 2 }$
E)2
F)-4
G)4
H)-2

Question

Find the value of the limit $\lim _ { x \rightarrow 1 } \frac { x - 1 } { \sqrt { x } - 1 }$
A)
$\sqrt { 2 }$ 
B)1
C)-1
D) $- \sqrt { 2 }$
E)2
F)-4
G)4
H)-2

Quizplus · Accepted Answer

The limit can be evaluated by rationalizing the denominator. Multiply the numerator and the denominator by the conjugate of the denominator, $\sqrt{x} + 1$. This gives $\lim_{x \rightarrow 1} \frac{(x - 1)(\sqrt{x} + 1)}{x - 1}$. The $x - 1$ terms cancel out, leaving $\lim_{x \rightarrow 1} (\sqrt{x} + 1)$, which evaluates to $2$ as $x$ approaches $1$.

Find the Value of the Limit lim⁡x→1x−1x−1\lim _ { x \rightarrow 1 } \frac { x - 1 } { \sqrt { x } - 1 }limx→1​x​−1x−1​

Find the Value of the Limit $\lim _ { x \rightarrow 1 } \frac { x - 1 } { \sqrt { x } - 1 }$