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Question
\[\lim_{x\to3}\frac{(84-x)^{\frac{1}{4}}-3}{x-3}\mathrm{~is}\]
Options
\[\frac{-1}{108}\]
\[\frac{-1}{84}\]
\[\frac{-1}{27}\]
\[\frac{-1}{4}\]
MCQ
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Solution
\[\frac{-1}{108}\]
Explanation:
\[\lim_{x\to3}\frac{(84-x)^{\frac{1}{4}}-3}{x-3}\mathrm{~is}\]
Let \[(84-x)^{\frac{1}{4}}=\mathbf{t}\]
\[\Rightarrow(84-x)=\mathrm{t}^4\]
\[\Rightarrow x=84-\mathrm{t}^{4}\]
As x → 3 then t → 3
\[\therefore\quad\lim_{x\to3}\frac{(84-x)^{\frac{1}{4}}-3}{x-3}=\lim_{\mathrm{t\to3}}\frac{\mathrm{t-3}}{84-\mathrm{t}^{4}-3}\]
\[=\lim_{\mathrm{t\to3}}\frac{\mathrm{t-3}}{81-\mathrm{t}^{4}}\]
\[=\lim_{t\to0}\frac{-1}{\frac{(t^4-81)}{t-3}}\]
\[=\frac{-1}{4\times\left(3\right)^{3}}=\frac{-1}{108}\]
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