Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (1-3*x^2+2*x^3)/(x^3+2*x+4*x^2)
-
Limit of (-4-7*x+2*x^2)/(4-13*x+3*x^2)
-
Limit of 5+3*n
-
Limit of (-12+x^2-4*x)/(48+x^2-14*x)
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Identical expressions
- log(sin(x))/log(x)
- logarithm of ( sinus of (x)) divide by logarithm of (x)
- logsinx/logx
- log(sin(x)) divide by log(x)
-
Similar expressions
- log(sinx)/log(x)
Limit of the function log(sin(x))/log(x)
The solution
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[src]
/log(sin(x))\ lim |-----------| x->0+\ log(x) /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
Limit(log(sin(x))/log(x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x \right)}} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}{x \log{\left(x \right)}^{2} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}\right)$$
=
$$1$$
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
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x \right)}} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}{x \log{\left(x \right)}^{2} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}\right)$$
=
$$1$$
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]
/log(sin(x))\ lim |-----------| x->0+\ log(x) /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
1
$$1$$
= 1.00000000290572
/log(sin(x))\ lim |-----------| x->0-\ log(x) /
$$\lim_{x \to 0^-}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right)$$
1
$$1$$
= (1.00000000245513 + 1.02257373335904e-9j)
= (1.00000000245513 + 1.02257373335904e-9j)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = \infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = -\infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = \infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = -\infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(x \right)}}\right) = 0$$
More at x→-oo
The graph