How I can to continue this limit \lim_{xto0^+}\frac{1}{x}-\frac{1}{x^2}??

My limit is \lim_{xto0^+}\frac{1}{x}-\frac{1}{x^2}.

I make: \lim_{xto0^+}\frac{1}{x}-\frac{1}{x^2} = \lim_{xto0^+}\frac{x^2 - x}{x^3} = ?

I don't know How I can to continue simplification. Please, I need to resolve without the concept of derivative.

1 Answer
Apr 11, 2018

-oo

Explanation:

Rather than subtracting them, we may rewrite using negative exponents and simplify:

lim_(x->0^+)x^-1-x^-2=lim_(x->0^+)x^-1(1-x^-1)

Evaluate:

lim_(x->0^+)x^-1(1-x^-1)=oo(1-oo)=oo(-oo)=-oo. Multiplying a very large, positive number by a very large, negative number yields a very large, negative number.

As lim_(x->0^+)x^-1=lim_(x->0^+)1/x=oo

No differentiation needed whatsoever -- in fact, it would be wrong to apply l'Hospital's Rule, as we did not run into an indeterminate form with this simplification.