Evaluate the following limit : lim(x→a) (a sin x - x sin a)/ax^2 - xa^2

Question

Evaluate the following limit : \(\lim\limits_{\text x \to a}\cfrac{a\,sin\,\text x-\text x\,sin\,a}{a\text x^2-\text x a^2} \)

lim(x→a) (a sin x - x sin a)/ax² - xa²

Answer 1

Given, \(\lim\limits_{\text x \to a}\cfrac{a\,sin\,\text x-\text x\,sin\,a}{a\text x^2-\text x a^2} \)

Let t = x - a

Then, as x→a, t→0

Hence, \(\lim\limits_{\text x \to a}\cfrac{a\,sin\,\text x-\text x\,sin\,a}{a\text x^2-\text x a^2} \) = \(\cfrac{a\,cos\,a-sin\,a}{a^2}\)