Question

# Find the value limx→4(√(x−3−1)/x−4)

Functions
Find the value $$\displaystyle\lim{x}→{4}{\left(√\frac{{{x}−{3}−{1}}}{{x}}−{4}\right)}$$

$$\displaystyle\lim{x}\to{4}{\left({\left(\sqrt{{x}}-{3}\right)}-\frac{{1}}{{{x}-{4}}}\right)}$$
$$((\sqrt x-3)-1)/(x-4)=((\sqrt x-3)-1(\sqrt x-3)+1)/((x-4)(\sqrt x-3)+1) =(((\sqrt x-3)^2)-1^2)/((x-4)(\sqrt x-3)+1) =(x-4)/((x-4)(\sqrt x-3)+1) =1/(\sqrt x-3)+1$$
Therefore we have $$\displaystyle\lim{x}\to{4}\frac{{{\left(\sqrt{{x}}-{3}\right)}-{1}}}{{{x}-{4}}}=\lim{x}\to{4}\frac{{{\left(\sqrt{{x}}-{3}\right)}-{1}}}{{{x}-{4}}}=\lim{x}\to{4}{\left(\frac{{1}}{{\sqrt{{x}}-{3}}}-{1}\right)}=\frac{{1}}{{\sqrt{{4}}-{3}}}+{1}=\frac{{1}}{{2}}$$