Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (1-4*x)^(1/x)
-
Limit of (-16+x^2+6*x)/(-2-5*x+3*x^2)
-
Limit of (1+x)^(2/3)-(-1+x)^(2/3)
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Limit of 1/3+x/3
- Derivative of:
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x^5*log(x)
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Identical expressions
- x^ five *log(x)
- x to the power of 5 multiply by logarithm of (x)
- x to the power of five multiply by logarithm of (x)
- x5*log(x)
- x5*logx
- x⁵*log(x)
- x^5log(x)
- x5log(x)
- x5logx
- x^5logx
Limit of the function x^5*log(x)
The solution
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[src]
/ 5 \ lim \x *log(x)/ x->0+
$$\lim_{x \to 0^+}\left(x^{5} \log{\left(x \right)}\right)$$
Limit(x^5*log(x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} x^{5} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(x^{5} \log{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x^{5}}{\frac{d}{d x} \frac{1}{\log{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- 5 x^{5} \log{\left(x \right)}^{2}\right)$$
=
$$\lim_{x \to 0^+}\left(- 5 x^{5} \log{\left(x \right)}^{2}\right)$$
=
$$0$$
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
$$\lim_{x \to 0^+} x^{5} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(x^{5} \log{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x^{5}}{\frac{d}{d x} \frac{1}{\log{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- 5 x^{5} \log{\left(x \right)}^{2}\right)$$
=
$$\lim_{x \to 0^+}\left(- 5 x^{5} \log{\left(x \right)}^{2}\right)$$
=
$$0$$
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]
/ 5 \ lim \x *log(x)/ x->0+
$$\lim_{x \to 0^+}\left(x^{5} \log{\left(x \right)}\right)$$
0
$$0$$
= -8.59923486532194e-16
/ 5 \ lim \x *log(x)/ x->0-
$$\lim_{x \to 0^-}\left(x^{5} \log{\left(x \right)}\right)$$
0
$$0$$
= (4.10989734351127e-16 - 2.07295403913472e-16j)
= (4.10989734351127e-16 - 2.07295403913472e-16j)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(x^{5} \log{\left(x \right)}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{5} \log{\left(x \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(x^{5} \log{\left(x \right)}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(x^{5} \log{\left(x \right)}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{5} \log{\left(x \right)}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{5} \log{\left(x \right)}\right) = -\infty$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{5} \log{\left(x \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(x^{5} \log{\left(x \right)}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(x^{5} \log{\left(x \right)}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{5} \log{\left(x \right)}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{5} \log{\left(x \right)}\right) = -\infty$$
More at x→-oo
The graph