Use Limits to calculate slope of the tangent

Question

Use limits to calculate the slope of the tangent to the curve $y=\frac1x$ at $x=a$. I need to write an equation for the tangent to $y=\frac1x$ at $x=4$.

I think I understand the basics of the question and using the formula

$$\lim_ {a\to0}\frac {f(x+a)-f(x)}{a} $$

when I worked it out the way I think I suppose to I got

$\frac1{x(a+x)} $ then $\frac1{2(a+2)} \to \frac1{2(0+2)} \to \frac14 $

So is my limit $1/4$? How should my answer be?

Any help would be greatly appreciated. Thank you.

Answer 1

It is $$\lim_{h->0}\frac{\frac{1}{a+h}-\frac{1}{a}}{h}$$ for $$h\ne 0,a\neq 0$$

Answer 2

\begin{align} \lim_{h\to 0}\frac{f (x+h)-f (x)}h&=\lim_{h\to 0}\frac{\frac1{x+h}-\frac1x}h\\ &=\lim_{h\to 0}\frac{-h}{h(x+h)x}\\ &=\lim_{h\to 0}\frac{-1}{(x+h)x}\\ &=\frac{-1}{x^2} \end{align}