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lim-x-1-x-1-p-1-x-1-q-1-




Question Number 121410 by john santu last updated on 07/Nov/20
 lim_(x→1) (((x)^(1/p) −1)/( (x)^(1/q) −1)) ?
$$\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\sqrt[{\mathrm{p}}]{\mathrm{x}}−\mathrm{1}}{\:\sqrt[{\mathrm{q}}]{\mathrm{x}}−\mathrm{1}}\:?\: \\ $$
Answered by TANMAY PANACEA last updated on 07/Nov/20
x=t^(pq)   lim_(t→1)  ((t^q −1)/(t^p −1))  lim_(t→1)  (((t^q −1)/(t−1))/((t^p −1)/(t−1)))=((q×(1)^(q−1) )/(p×(1)^(p−1) ))=(q/p)
$${x}={t}^{{pq}} \\ $$$${li}\underset{{t}\rightarrow\mathrm{1}} {{m}}\:\frac{{t}^{{q}} −\mathrm{1}}{{t}^{{p}} −\mathrm{1}} \\ $$$${li}\underset{{t}\rightarrow\mathrm{1}} {{m}}\:\frac{\frac{{t}^{{q}} −\mathrm{1}}{{t}−\mathrm{1}}}{\frac{{t}^{{p}} −\mathrm{1}}{{t}−\mathrm{1}}}=\frac{{q}×\left(\mathrm{1}\right)^{{q}−\mathrm{1}} }{{p}×\left(\mathrm{1}\right)^{{p}−\mathrm{1}} }=\frac{{q}}{{p}} \\ $$
Answered by liberty last updated on 07/Nov/20
  lim_(x→1)  ((x^(1/p) −1)/(x^(1/q) −1)) = lim_(x→1)  (((1/p)x^((1/p)−1) )/((1/q)x^((1/q)−1) ))   lim_(x→1)  ((qx^((1−p)/p) )/(px^((1−q)/q) )) = (q/p).
$$\:\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{p}}} −\mathrm{1}}{\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{q}}} −\mathrm{1}}\:=\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{\mathrm{p}}\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{p}}−\mathrm{1}} }{\frac{\mathrm{1}}{\mathrm{q}}\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{q}}−\mathrm{1}} } \\ $$$$\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{qx}^{\frac{\mathrm{1}−\mathrm{p}}{\mathrm{p}}} }{\mathrm{px}^{\frac{\mathrm{1}−\mathrm{q}}{\mathrm{q}}} }\:=\:\frac{\mathrm{q}}{\mathrm{p}}. \\ $$

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