Question

f(x) = 1/x, f(x)-f(a) / x-a

Answer 1

\[ \frac{ f(x+a) - f(x)}{(x+a)-x}
\]

Answer 2

\[\frac{ \frac{1}{x+a} - \frac{1}{x} }{a} \\ \frac{1}{a}\left( \frac{1}{x+a} - \frac{1}{x}\right)\]
now use a common denominator of x(x+a)
\[ \frac{1}{a}\left( \frac{x}{x(x+a)} - \frac{x+a}{x(x+a)}\right) \\
\frac{1}{a}\left( \frac{x-x-a}{x(x+a)} \right) \\
\frac{1}{\cancel{a}}\left( \frac{-\cancel{a}}{x(x+a)} \right) \\
- \frac{1}{x(x+a)}
\]

now take the limit, and let a-->0
\[\lim_{a \rightarrow 0} \frac{ -1 }{ x(x+a) }= \frac{ -1 }{ x^2}= -x^{-2}\]

Answer 3

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