Limit with Weird Behaviour

Calculus Level 1

$\large \lim_{x\to 0} x^{\frac{0}{x}}$

Let $x \in \mathbb{R}$ , then what is the value of the above expression?

Bonus: What happens if we allow for $x \in \mathbb{C}$ ?

-1 0 1

e

Limit does not exist

4 solutions

Sharky Kesa
Apr 25, 2017

Well, for the question, we simplify $\frac{0}{x}=0$ , and then we're left with $\displaystyle \lim_{x \to 0} x^0=1$ , so that's done.

Let's start the discussion on the behaviour of the function in more general systems.

this is actually wrong as i thought, when x = 0 the limit can't exist on real line

aaaaaaa aaaaaa - 4 years ago

0 to the power 0 is an undefined term, you cannot say it equals 1.

Ashkar Awal - 4 years ago

But this is a limit as $x$ approaches 0. We are not evaluating $0^0$ .

Sharky Kesa - 4 years ago

Ya I agree with sharks as we are not evaluating 0^0as it is not given in question thatx=0

Biswajit Barik - 3 years, 11 months ago

Maybe this.

$\displaystyle \lim _ { x \to 0 + }x ^ { \frac { 0 } { x } } = \lim _ { x \to 0 - } x ^ { \frac { 0 } { x } } = 1$ .

. . - 3 weeks ago

. .
May 21, 2021

Just substitute a number which is near to zero.

Like $\displaystyle \lim _ { x \to 0 } x ^ { \frac { 0 } { x } } \rightarrow 1$ .

Bonus : if $x \in \mathbb { C }$ , then it is also $1$ because if any number in complex number exceeding zero, then $\displaystyle \frac { 0 } { x } = 0$ . Then it is always $x ^ { 0 } = 1 \because x \ne 0$ .

. . - 3 weeks ago

Nikhil Kumar
May 27, 2017

Let y = lim x^(0/x)

Taking log on both sides;

Log y = lim (0/x) log x; where x->0

log y = lim( 0 * log x) where x->0

log y = 0

=> y = 1. Answer.

Rafiq Haq
May 27, 2017

Seems to be some alternative possibilities for the limiting value according to Wolfram Alpha

Limit with Weird Behaviour

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