lim_(x->a) (x^(9/2)-a^(9/2))/(x^(1/2)-a^(1/2))=? 

প্রশ্ন: \( \lim_{x \to a} \frac{x^{9/2} - a^{9/2}}{x^{1/2} - a^{1/2}} = ? \)

সমাধান:

ধরি, \( x^{1/2} = u \) এবং \( a^{1/2} = b \). তাহলে, \( x = u^2 \) এবং \( a = b^2 \).

সুতরাং, \( x \to a \) হলে, \( u \to b \) হবে।

অতএব,

\( \lim_{x \to a} \frac{x^{9/2} - a^{9/2}}{x^{1/2} - a^{1/2}} = \lim_{u \to b} \frac{(u^2)^{9/2} - (b^2)^{9/2}}{u - b} = \lim_{u \to b} \frac{u^9 - b^9}{u - b} \)

আমরা জানি, \( \lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1} \).

সুতরাং, \( \lim_{u \to b} \frac{u^9 - b^9}{u - b} = 9b^{9-1} = 9b^8 \).

যেহেতু \( b = a^{1/2} \), তাই \( 9b^8 = 9(a^{1/2})^8 = 9a^{8/2} = 9a^4 \).

অতএব, \( \lim_{x \to a} \frac{x^{9/2} - a^{9/2}}{x^{1/2} - a^{1/2}} = 9a^4 \). 🎉