I am trying to solve $\underset{x \to +6}\lim \frac{\frac{1}{-6}+\frac{1}{x}}{-6+x}$ without L'hospital's Rule.
I don't understand why the website I'm working with didn't just write $\underset{x \to +6}\lim \frac{\frac{1}{-6}+\frac{1}{x}}{-6+x}$ as $\underset{x \to +6}\lim \frac{\frac{-1}{6}+\frac{1}{x}}{x-6}$, but I have checked on Desmos, and these seem equivalent.
$$\underset{x \to +6}\lim \frac{\frac{-1}{6}+\frac{1}{x}}{x-6}$$
$$=\underset{x \to +6}\lim \frac{6-x}{6x(x-6)}$$
$$=\underset{x \to +6}\lim \frac{6-x}{6x^2-36x}$$
$$=\underset{x \to +6}\lim\frac{6-(6)}{6(6)^2-36(6)}$$
$$=\underset{x \to +6}\lim\frac{0}{2126-2126}$$
I have also done other calculations with it, but I can't seem to find a way to solve it. Could someone point me in the right direction?
In your second step, isolate - sign from the numerator. Then $(x-6)$ gets cancelled.
And we finally have,
$$\underset{x \to +6}\lim \frac{-1}{6x} = -\frac{1}{36}$$