Calculate: \lim_{x arrow 1} ((sqrt(x-1)+2x-2)/(4x^2-4))

Expression: $\lim_{x \rightarrow 1} \left(\frac{ \sqrt{ x-1 }+2x-2 }{ 4{x}^{2}-4 }\right)$

Since the function $\frac{ \sqrt{ x-1 }+2x-2 }{ 4{x}^{2}-4 }$ is undefined on the left side of $1$, evaluate the right-hand limit

$\lim_{x \rightarrow 1^+} \left(\frac{ \sqrt{ x-1 }+2x-2 }{ 4{x}^{2}-4 }\right)$

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\lim_{x \rightarrow 1^+} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( \sqrt{ x-1 }+2x-2 \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( 4{x}^{2}-4 \right) }\right)$

Find the derivative

$\lim_{x \rightarrow 1^+} \left(\frac{ \frac{ 1 }{ 2\sqrt{ x-1 } }+2 }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( 4{x}^{2}-4 \right) }\right)$

Find the derivative

$\lim_{x \rightarrow 1^+} \left(\frac{ \frac{ 1 }{ 2\sqrt{ x-1 } }+2 }{ 8x }\right)$

Write all numerators above the common denominator

$\lim_{x \rightarrow 1^+} \left(\frac{ \frac{ 1+4\sqrt{ x-1 } }{ 2\sqrt{ x-1 } } }{ 8x }\right)$

Simplify the complex fraction

$\lim_{x \rightarrow 1^+} \left(\frac{ 1+4\sqrt{ x-1 } }{ 16x\sqrt{ x-1 } }\right)$

Separate the fraction into $2$ fractions

$\lim_{x \rightarrow 1^+} \left(\frac{ 1+4\sqrt{ x-1 } }{ 16x } \times \frac{ 1 }{ \sqrt{ x-1 } }\right)$

Evaluate the limit of each term separately

$\begin{array} { l }\lim_{x \rightarrow 1^+} \left(\frac{ 1+4\sqrt{ x-1 } }{ 16x }\right),\\\lim_{x \rightarrow 1^+} \left(\frac{ 1 }{ \sqrt{ x-1 } }\right)\end{array}$

Evaluate the limit

$\begin{array} { l }\frac{ 1 }{ 16 },\\\lim_{x \rightarrow 1^+} \left(\frac{ 1 }{ \sqrt{ x-1 } }\right)\end{array}$

Evaluate the limit

$\begin{array} { l }\frac{ 1 }{ 16 },\\+\infty\end{array}$

Since the expression $\begin{array} { l }a \times \left( +\infty \right),& a > 0\end{array}$ is defined as $+\infty$, the limit $\lim_{x \rightarrow 1^+} \left(\frac{ 1+4\sqrt{ x-1 } }{ 16x } \times \frac{ 1 }{ \sqrt{ x-1 } }\right)$ equals $+\infty$

$+\infty$