Question

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.

Answer 1

`-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.