Solve for: \lim_{x arrow 0} ((sqrt(1+x)-1)/(sqrt(1-x)-1))

Expression: $\lim_{x \rightarrow 0} \left(\frac{ \sqrt{ 1+x }-1 }{ \sqrt{ 1-x }-1 }\right)$

Evaluate the limits of numerator and denominator separately

$\begin{array} { l }\lim_{x \rightarrow 0} \left(\sqrt{ 1+x }-1\right),\\\lim_{x \rightarrow 0} \left(\sqrt{ 1-x }-1\right)\end{array}$

Evaluate the limit

$\begin{array} { l }0,\\\lim_{x \rightarrow 0} \left(\sqrt{ 1-x }-1\right)\end{array}$

Evaluate the limit

$\begin{array} { l }0,\\0\end{array}$

Since the expression $\frac{ 0 }{ 0 }$ is an indeterminate form, try transforming the expression

$\lim_{x \rightarrow 0} \left(\frac{ \sqrt{ 1+x }-1 }{ \sqrt{ 1-x }-1 }\right)$

Rationalize the denominator

$\lim_{x \rightarrow 0} \left(\frac{ \left( \sqrt{ 1+x }-1 \right) \times \left( \sqrt{ 1-x }+1 \right) }{ -x }\right)$

Simplify the expression

$\lim_{x \rightarrow 0} \left(\frac{ \sqrt{ \left( 1+x \right) \times \left( 1-x \right) }+\sqrt{ 1+x }-\sqrt{ 1-x }-1 }{ -x }\right)$

Use $\left( a-b \right)\left( a+b \right)={a}^{2}-{b}^{2}$ to simplify the product

$\lim_{x \rightarrow 0} \left(\frac{ \sqrt{ 1-{x}^{2} }+\sqrt{ 1+x }-\sqrt{ 1-x }-1 }{ -x }\right)$

Use $\frac{ -a }{ b }=\frac{ a }{ -b }=-\frac{ a }{ b }$ to rewrite the fraction

$\lim_{x \rightarrow 0} \left(-\frac{ \sqrt{ 1-{x}^{2} }+\sqrt{ 1+x }-\sqrt{ 1-x }-1 }{ x }\right)$

Since the function $-\frac{ \sqrt{ 1-{x}^{2} }+\sqrt{ 1+x }-\sqrt{ 1-x }-1 }{ x }$ is undefined for $0$, evaluate the left-hand and right-hand limits

$\begin{array} { l }\lim_{x \rightarrow 0^-} \left(-\frac{ \sqrt{ 1-{x}^{2} }+\sqrt{ 1+x }-\sqrt{ 1-x }-1 }{ x }\right),\\\lim_{x \rightarrow 0^+} \left(-\frac{ \sqrt{ 1-{x}^{2} }+\sqrt{ 1+x }-\sqrt{ 1-x }-1 }{ x }\right)\end{array}$

Evaluate the limit

$\begin{array} { l }-1,\\\lim_{x \rightarrow 0^+} \left(-\frac{ \sqrt{ 1-{x}^{2} }+\sqrt{ 1+x }-\sqrt{ 1-x }-1 }{ x }\right)\end{array}$

Evaluate the limit

$\begin{array} { l }-1,\\-1\end{array}$

Since the left-hand and right-hand limits are equal, the limit $\lim_{x \rightarrow 0} \left(-\frac{ \sqrt{ 1-{x}^{2} }+\sqrt{ 1+x }-\sqrt{ 1-x }-1 }{ x }\right)$ equals $-1$

$-1$