Mister Exam
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How to use it?
- Limit of the function:
-
Limit of (-exp(-x)+exp(x))/(exp(x)+exp(-x))
-
Limit of (3+3*x^3+5*x+5*x^2)/(-1+x^2)
-
Limit of (3-x^2+5*x)/(4*x^7+81*x)
-
Limit of -cos(1/x)+2*x*sin(1/x)
- Derivative of:
-
x^4*log(x)
- Integral of d{x}:
- x^4*log(x)
-
Identical expressions
- x^ four *log(x)
- x to the power of 4 multiply by logarithm of (x)
- x to the power of four multiply by logarithm of (x)
- x4*log(x)
- x4*logx
- x⁴*log(x)
- x^4log(x)
- x4log(x)
- x4logx
- x^4logx
Limit of the function x^4*log(x)
The solution
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[src]
/ 4 \ lim \x *log(x)/ x->0+
$$\lim_{x \to 0^+}\left(x^{4} \log{\left(x \right)}\right)$$
Limit(x^4*log(x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} x^{4} = 0$$
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.
$$\lim_{x \to 0^+}\left(x^{4} \log{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x^{4}}{\frac{d}{d x} \frac{1}{\log{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- 4 x^{4} \log{\left(x \right)}^{2}\right)$$
=
$$\lim_{x \to 0^+}\left(- 4 x^{4} \log{\left(x \right)}^{2}\right)$$
=
$$0$$
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
$$\lim_{x \to 0^+} x^{4} = 0$$
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.
$$\lim_{x \to 0^+}\left(x^{4} \log{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x^{4}}{\frac{d}{d x} \frac{1}{\log{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- 4 x^{4} \log{\left(x \right)}^{2}\right)$$
=
$$\lim_{x \to 0^+}\left(- 4 x^{4} \log{\left(x \right)}^{2}\right)$$
=
$$0$$
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]
/ 4 \ lim \x *log(x)/ x->0+
$$\lim_{x \to 0^+}\left(x^{4} \log{\left(x \right)}\right)$$
0
$$0$$
= -4.17440384416252e-13
/ 4 \ lim \x *log(x)/ x->0-
$$\lim_{x \to 0^-}\left(x^{4} \log{\left(x \right)}\right)$$
0
$$0$$
= (-3.92017193388019e-13 + 2.09170932732045e-13j)
= (-3.92017193388019e-13 + 2.09170932732045e-13j)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(x^{4} \log{\left(x \right)}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{4} \log{\left(x \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(x^{4} \log{\left(x \right)}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(x^{4} \log{\left(x \right)}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{4} \log{\left(x \right)}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{4} \log{\left(x \right)}\right) = \infty$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{4} \log{\left(x \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(x^{4} \log{\left(x \right)}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(x^{4} \log{\left(x \right)}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{4} \log{\left(x \right)}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{4} \log{\left(x \right)}\right) = \infty$$
More at x→-oo
The graph