IGNOU MTE 1 2026 - ) ज्ञात कीजिए।10. a) यदि है, तो दर्शाइए कि :

Question

) $\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$ ज्ञात कीजिए।
10. a) यदि $I_{m,n} = \int x^3 (\log x)^n \, dx$ है, तो दर्शाइए कि :

09 Jan 2026

Answer :

Word Count : 205

पहले सीमा को हल करते हैं: [ \lim_{x \to 0} \frac{\tan x - \sin x}{x^3}. ] चरण 1: टेलर श्रृंखला का विस्तार * (\tan x = x + \frac{x^3}{3} + O(x^5)) * (\sin x = x - \frac{x^3}{6} + O(x^5)) इसलिए: [ \tan x - \sin x = \left(x + \frac{x^3}{3} + \dots \right) - \left(x - \frac{x^3}{6} + \dots \right) = \frac{x^3}{3} + \frac{x^3}{6} + \dots = \frac{x^3}{2} + O(x^5) ] चरण 2:

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