Evaluate: \lim_{x arrow 2} ((x-2)/(\frac{1){x}-(1)/(2)})

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

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

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

Find the derivative

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

Find the derivative

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

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

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

Simplify the complex fraction

$\lim_{x \rightarrow 2} \left(-{x}^{2}\right)$

Evaluate the limit

$-{2}^{2}$

Evaluate the power

$-4$

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