Question

Find the value limx→4(√(x−3−1)/x−4)

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asked 2021-02-15
Find the value \(\displaystyle\lim{x}→{4}{\left(√\frac{{{x}−{3}−{1}}}{{x}}−{4}\right)}\)

Answers (1)

2021-02-16

We have to find
\(\displaystyle\lim{x}\to{4}{\left({\left(\sqrt{{x}}-{3}\right)}-\frac{{1}}{{{x}-{4}}}\right)}\)
Note that we can not find the value of limit by plugging x=4, as in that case the denominator become zero. Let us first rationalize numerator.
\(((\sqrt x-3)-1)/(x-4)=((\sqrt x-3)-1(\sqrt x-3)+1)/((x-4)(\sqrt x-3)+1) =(((\sqrt x-3)^2)-1^2)/((x-4)(\sqrt x-3)+1) =(x-4)/((x-4)(\sqrt x-3)+1) =1/(\sqrt x-3)+1\)
Therefore we have \(\displaystyle\lim{x}\to{4}\frac{{{\left(\sqrt{{x}}-{3}\right)}-{1}}}{{{x}-{4}}}=\lim{x}\to{4}\frac{{{\left(\sqrt{{x}}-{3}\right)}-{1}}}{{{x}-{4}}}=\lim{x}\to{4}{\left(\frac{{1}}{{\sqrt{{x}}-{3}}}-{1}\right)}=\frac{{1}}{{\sqrt{{4}}-{3}}}+{1}=\frac{{1}}{{2}}\)

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