Lim x⇒0 [tex]\sqrt{1+x}[/tex] -1 / [tex]x^{2}[/tex]

Предыдущий
вопрос Следующий
вопрос

To find the limit as x approaches 0 of the given expression:

lim x→0 (√1+x - 1) / x^2

First, let's simplify the expression (√1+x - 1) / x^2 by rationalizing the numerator.
Multiply by the conjugate of the numerator:

= lim x→0 [(√1+x - 1) / x^2] * [(√1+x + 1) / (√1+x + 1)]
= lim x→0 [(1+x - 1) / (x^2(√1+x + 1))]
= lim x→0 [(x) / (x^2(√1+x + 1))]
= lim x→0 [1 / (x(√1+x + 1))]

Now we can attempt to find the limit as x approaches 0:
As x approaches 0, the expression tends to infinity, therefore the limit does not exist.

Therefore, lim x→0 (√1+x - 1) / x^2 does not exist.