Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (9+3*x^2+4*x)/(7-7*x+3*x^2)
-
Limit of (1+2*n)/(-1+3*n)
-
Limit of (x^3-3^x)/(-3+x)
-
Limit of -x+(-2+x)^4/(3+x)^4
-
Identical expressions
- x^(x^x)/x
- x to the power of (x to the power of x) divide by x
- x(xx)/x
- xxx/x
- x^x^x/x
- x^(x^x) divide by x
Limit of the function x^(x^x)/x
The solution
You have entered
[src]
/ / x\\
| \x /|
|x |
lim |-----|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{x^{x^{x}}}{x}\right)$$
Limit(x^(x^x)/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} x^{x^{x}} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{x^{x^{x}}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x^{x^{x}}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(x^{x^{x}} \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x}}{x}\right)\right)$$
=
$$\lim_{x \to 0^+}\left(x^{x^{x}} \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x}}{x}\right)\right)$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} x^{x^{x}} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{x^{x^{x}}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x^{x^{x}}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(x^{x^{x}} \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x}}{x}\right)\right)$$
=
$$\lim_{x \to 0^+}\left(x^{x^{x}} \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x}}{x}\right)\right)$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
One‐sided limits
[src]
/ / x\\
| \x /|
|x |
lim |-----|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{x^{x^{x}}}{x}\right)$$
1
$$1$$
= 1.0151367575031
/ / x\\
| \x /|
|x |
lim |-----|
x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{x^{x^{x}}}{x}\right)$$
1
$$1$$
= (0.986980095563771 + 0.0121900932593533j)
= (0.986980095563771 + 0.0121900932593533j)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{x^{x^{x}}}{x}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x^{x^{x}}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{x^{x^{x}}}{x}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{x^{x^{x}}}{x}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x^{x^{x}}}{x}\right) = 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x^{x^{x}}}{x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x^{x^{x}}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{x^{x^{x}}}{x}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{x^{x^{x}}}{x}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x^{x^{x}}}{x}\right) = 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x^{x^{x}}}{x}\right) = 0$$
More at x→-oo
The graph