Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow7}\frac{4-\sqrt{9+\text{x}}}{1-\sqrt{8-\text{x}}}$
$\lim\limits_{\text{x}\rightarrow7}\frac{4-\sqrt{9+\text{x}}}{1-\sqrt{8-\text{x}}}$
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Answer
$\lim\limits_{\text{x}\rightarrow7}\frac{4-\sqrt{9+\text{x}}}{1-\sqrt{8-\text{x}}}$
$=\lim\limits_{\text{x}\rightarrow7}\frac{\big(4-\sqrt{9+\text{x}}\big)}{1-\sqrt{8-\text{x}}}\times\frac{\big(4+\sqrt{9+\text{x}}\big)}{\big(4+\sqrt{9+\text{x}}\big)}\times\frac{\big(1+\sqrt{8-\text{x}}\big)}{\big(\sqrt{1+\sqrt{8+\text{x}}}\big)}$
$=\lim\limits_{\text{x}\rightarrow7}\frac{\big((4)^2-\big(\sqrt{9+\text{x}}\big)^2\big)}{\big((1)^2-\big(\sqrt{8-\text{x}}\big)^2\big)}\times\frac{1+\sqrt{8-\text{x}}}{4+\sqrt{9+\text{x}}}$
$=\lim\limits_{\text{x}\rightarrow7}\frac{(16-9-\text{x})\times\big(1+\sqrt{8-\text{x}}\big)}{(1-8+\text{x})\times\big(4+\sqrt{9+\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow7}\frac{7-\text{x}}{(-7)(7-\text{x})}\frac{\big(1+\sqrt{8-\text{x}}\big)}{\big(4+\sqrt{9+\text{x}}\big)}$
$=\frac{1}{(-1)}\times\frac{(1+1)}{(4+4)}=\frac{-1}{4}$
$=\lim\limits_{\text{x}\rightarrow7}\frac{\big(4-\sqrt{9+\text{x}}\big)}{1-\sqrt{8-\text{x}}}\times\frac{\big(4+\sqrt{9+\text{x}}\big)}{\big(4+\sqrt{9+\text{x}}\big)}\times\frac{\big(1+\sqrt{8-\text{x}}\big)}{\big(\sqrt{1+\sqrt{8+\text{x}}}\big)}$
$=\lim\limits_{\text{x}\rightarrow7}\frac{\big((4)^2-\big(\sqrt{9+\text{x}}\big)^2\big)}{\big((1)^2-\big(\sqrt{8-\text{x}}\big)^2\big)}\times\frac{1+\sqrt{8-\text{x}}}{4+\sqrt{9+\text{x}}}$
$=\lim\limits_{\text{x}\rightarrow7}\frac{(16-9-\text{x})\times\big(1+\sqrt{8-\text{x}}\big)}{(1-8+\text{x})\times\big(4+\sqrt{9+\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow7}\frac{7-\text{x}}{(-7)(7-\text{x})}\frac{\big(1+\sqrt{8-\text{x}}\big)}{\big(4+\sqrt{9+\text{x}}\big)}$
$=\frac{1}{(-1)}\times\frac{(1+1)}{(4+4)}=\frac{-1}{4}$
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