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Question
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
Solution
\[\lim_{x \to a} \frac{xf\left( a \right) - af\left( x \right)}{x - a}\]
\[ = \lim_{x \to a} \frac{xf\left( a \right) - af\left( x \right) - xf\left( x \right) + xf\left( x \right)}{x - a}\]
\[ = \lim_{x \to a} \frac{xf\left( a \right) - xf\left( x \right) + xf\left( x \right) - af\left( x \right)}{x - a}\]
\[ = \lim_{x \to a} \frac{- x\left( f\left( x \right) - f\left( a \right) \right) + \left( x - a \right)f\left( x \right)}{x - a}\]
\[ = \lim_{x \to a} - x \lim_{x \to a} \frac{f\left( x \right) - f\left( a \right)}{x - a} + \lim_{x \to a} \frac{\left( x - a \right)f\left( x \right)}{x - a}\]
\[ = - a f'\left( a \right) + f(a)\]
\[\]
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