Question

\[\lim_{x\to3}\frac{(84-x)^{\frac{1}{4}}-3}{x-3}\mathrm{~is}\]

पर्याय

• \[\frac{-1}{108}\]

• \[\frac{-1}{84}\]

• \[\frac{-1}{27}\]

• \[\frac{-1}{4}\]

Answer 1

\[\frac{-1}{108}\]

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

\[\therefore\quad\lim_{x\to3}\frac{(84-x)^{\frac{1}{4}}-3}{x-3}=\lim_{\mathrm{t\to3}}\frac{\mathrm{t-3}}{84-\mathrm{t}^{4}-3}\]

\[=\lim_{\mathrm{t\to3}}\frac{\mathrm{t-3}}{81-\mathrm{t}^{4}}\]

\[=\lim_{t\to0}\frac{-1}{\frac{(t^4-81)}{t-3}}\]

\[=\frac{-1}{4\times\left(3\right)^{3}}=\frac{-1}{108}\]