Question

Limit to be evaluated without L'Hôpital's rule.
$\lim \limits_{x \to 0} (x \ln(x))$
​

Answer 1

lim In x

x-> 0

= - ∞ , which is does not exist

so, I simple language

f(x) = Inx

y = In x = log_e (x)

x = e^y

so x -> 0, e^y ->0

but, e^y > 0 for all y belongs to R

so as e^y -> 0 x -> - ∞

y = -∞

so lim In x = - ∞

x->0

Answer 2

Question:-

Limit to be evaluated without L'Hôpital's rule.

$\sf \large\lim \limits_{x \to 0} (x \ln(x)).$

Given:-

$\sf \large\lim \limits_{x \to 0} (x \ln(x)).$

To Show:-

• Limit to be evaluated without L'Hôpital's rule.

Solution:-

$\sf \large\lim \limits_{x \to 0} (x \ln(x))$

$\sf \large\lim \limits_{h \to 0} (0 + h)ln(0 + h).$

$\sf \large\lim \limits_{h \to 0} \: \: h(ln(h) = 0).$

$\sf \large\lim \limits_{x \to 0} (x \ln(x)).$

$\sf \large = \lim \limits_{h \to 0}(0 - h)(ln ( - h)) = 0.$

$\sf \large So, \lim \limits_{x \to 0} (x \ln(x)) = 0.$

Answer:-

$\sf \large \color{lime}\lim \limits_{x \to 0} (x \ln(x)) = 0.$

Hope you have satisfied. ⚘