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

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

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

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

Find the derivative

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

Find the derivative

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

Write all numerators above the common denominator

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

Simplify the expression

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

Any expression divided by $1$ remains the same

$\lim_{x \rightarrow 1} \left(4x\sqrt{ x }-1\right)$

Evaluate the limit

$4 \times 1\sqrt{ 1 }-1$

Simplify the expression

$3$

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