[latex]\begin{align}\underset{x\to 4}{\mathrm{lim}}\left(\frac{4-x}{\sqrt{x}-2}\right)&=\underset{x\to 4}{\mathrm{lim}}\left(\frac{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}{\sqrt{x}-2}\right)&& \text{Factor.} \\ &=\underset{x\to 4}{\mathrm{lim}}\left(\frac{\left(2+\sqrt{x}\right)\cancel{\left(2-\sqrt{x}\right)}}{-\cancel{\left(2-\sqrt{x}\right)}}\right)&& \text{Factor }-1\text{ out of the denominator and implify}. \\ &=\underset{x\to 4}{\mathrm{lim}}-\left(2+\sqrt{x}\right)&& \text{Evaluate}. \\ &=-\left(2+\sqrt{4}\right) \\ &=-4 \end{align}[/latex]

Analysis of the Solution

Multiplying by a conjugate would expand the numerator; look instead for factors in the numerator. Four is a perfect square so that the numerator is in the form

[latex]{a}^{2}-{b}^{2}[/latex]

and may be factored as

[latex]\left(a+b\right)\left(a-b\right)[/latex].

The function is undefined at [latex]x=7[/latex], so we will try values close to 7 from the left and the right.

Left-hand limit: [latex]\frac{|6.9 - 7|}{6.9 - 7}=\frac{|6.99 - 7|}{6.99 - 7}=\frac{|6.999 - 7|}{6.999 - 7}=-1[/latex]

Right-hand limit: [latex]\frac{|7.1 - 7|}{7.1 - 7}=\frac{|7.01 - 7|}{7.01 - 7}=\frac{|7.001 - 7|}{7.001 - 7}=1[/latex]

Since the left- and right-hand limits are not equal, there is no limit.