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
VNVDVI
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.

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