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$$\lim _{ x\rightarrow 4 }{ \frac { (\sqrt { 5-x } )-1 }{ 2-\sqrt { x } } } $$

Steps I took:

$$\lim _{ x\rightarrow 4 }{ \frac { (\sqrt { 5-x } )-1 }{ 2-\sqrt { x } } } \cdot \frac { (\sqrt { 5-x } )+1 }{ (\sqrt { 5-x } )+1 } $$

$$\lim _{ x\rightarrow 4 }{ \frac { 5-x-1 }{ (2-\sqrt { x } )(\sqrt { 5-x } +1) } } =\frac { -x+4 }{ 2\sqrt { 5-x } +2-\sqrt { x } \sqrt { 5-x } -\sqrt { x } } $$

At this point I don't know how else to manipulate it in order to get it to a form in which I could evaluate the limit. I would like hints rather than a direct answer.

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  • $\begingroup$ Divide numerator and denominator by x $\endgroup$ – Quality Jan 13 '15 at 4:07
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    $\begingroup$ Multiply also by $\frac{2+\sqrt{x}}{2+\sqrt{x}}$. Do not "simplify" unless it is a good idea. $\endgroup$ – André Nicolas Jan 13 '15 at 4:08
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From the step $$\lim _{ x\rightarrow 4 }{ \frac { 5-x-1 }{ (2-\sqrt { x } )(\sqrt { 5-x } +1) } } = \lim _{ x\rightarrow 4 }{ \frac { 4-x }{ (2-\sqrt { x } )(\sqrt { 5-x } +1) } }$$ Factor $$4-x=(2-\sqrt{x})(2+\sqrt{x})$$ then, $$\lim _{ x\rightarrow 4 }{ \frac { 4-x }{ (2-\sqrt { x } )(\sqrt { 5-x } +1) } }=\lim _{ x\rightarrow 4 }{ \frac { 2+\sqrt{x} }{ (\sqrt { 5-x } +1) } }=\frac{4}{2}=2.$$

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