Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
- Limit of (-1+a^x)/(-1+c^x)
-
Limit of atan(2*x)/(3*x)
-
Limit of atan(2*x)/(2*sin(x))
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Limit of asin(x)/(-3+x)
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Identical expressions
- atan(two *x)/(three *x)
- arc tangent of gent of (2 multiply by x) divide by (3 multiply by x)
- arc tangent of gent of (two multiply by x) divide by (three multiply by x)
- atan(2x)/(3x)
- atan2x/3x
- atan(2*x) divide by (3*x)
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Similar expressions
- atan(2*x)/(3*x^2)
- arctan(2*x)/(3*x)
Limit of the function atan(2*x)/(3*x)
The solution
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/atan(2*x)\ lim |---------| x->0+\ 3*x /
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right)$$
Limit(atan(2*x)/((3*x)), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right)$$
Do replacement
$$u = \operatorname{atan}{\left(2 x \right)}$$
$$x = \frac{\tan{\left(u \right)}}{2}$$
we get
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = \frac{\lim_{u \to 0^+}\left(\frac{\operatorname{atan}{\left(\frac{2 \tan{\left(u \right)}}{2} \right)}}{\frac{1}{2} \tan{\left(u \right)}}\right)}{3}$$
=
$$\frac{\lim_{u \to 0^+}\left(\frac{2 \operatorname{atan}{\left(\tan{\left(u \right)} \right)}}{\tan{\left(u \right)}}\right)}{3} = \frac{\lim_{u \to 0^+}\left(\frac{2 u}{\tan{\left(u \right)}}\right)}{3}$$
=
$$\frac{2 \lim_{u \to 0^+} \frac{1}{\frac{1}{u} \tan{\left(u \right)}}}{3}$$
transform
$$\lim_{u \to 0^+}\left(\frac{\tan{\left(u \right)}}{u}\right) = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u \cos{\left(u \right)}}\right)$$
=
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right) \lim_{u \to 0^+} \cos{\left(u \right)} = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
The limit
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
is first remarkable limit, is equal to 1.
The final answer:
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = \frac{2}{3}$$
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right)$$
Do replacement
$$u = \operatorname{atan}{\left(2 x \right)}$$
$$x = \frac{\tan{\left(u \right)}}{2}$$
we get
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = \frac{\lim_{u \to 0^+}\left(\frac{\operatorname{atan}{\left(\frac{2 \tan{\left(u \right)}}{2} \right)}}{\frac{1}{2} \tan{\left(u \right)}}\right)}{3}$$
=
$$\frac{\lim_{u \to 0^+}\left(\frac{2 \operatorname{atan}{\left(\tan{\left(u \right)} \right)}}{\tan{\left(u \right)}}\right)}{3} = \frac{\lim_{u \to 0^+}\left(\frac{2 u}{\tan{\left(u \right)}}\right)}{3}$$
=
$$\frac{2 \lim_{u \to 0^+} \frac{1}{\frac{1}{u} \tan{\left(u \right)}}}{3}$$
/tan(u)\
= 2/3 / ( lim |------| )
u->0+\ u / transform
$$\lim_{u \to 0^+}\left(\frac{\tan{\left(u \right)}}{u}\right) = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u \cos{\left(u \right)}}\right)$$
=
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right) \lim_{u \to 0^+} \cos{\left(u \right)} = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
The limit
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
is first remarkable limit, is equal to 1.
The final answer:
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = \frac{2}{3}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \operatorname{atan}{\left(2 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(3 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{atan}{\left(2 x \right)}}{\frac{d}{d x} 3 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2}{3 \left(4 x^{2} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+} \frac{2}{3}$$
=
$$\lim_{x \to 0^+} \frac{2}{3}$$
=
$$\frac{2}{3}$$
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
$$\lim_{x \to 0^+} \operatorname{atan}{\left(2 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(3 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{atan}{\left(2 x \right)}}{\frac{d}{d x} 3 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2}{3 \left(4 x^{2} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+} \frac{2}{3}$$
=
$$\lim_{x \to 0^+} \frac{2}{3}$$
=
$$\frac{2}{3}$$
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]
/atan(2*x)\ lim |---------| x->0+\ 3*x /
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right)$$
2/3
$$\frac{2}{3}$$
= 0.666666666666667
/atan(2*x)\ lim |---------| x->0-\ 3*x /
$$\lim_{x \to 0^-}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right)$$
2/3
$$\frac{2}{3}$$
= 0.666666666666667
= 0.666666666666667
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = \frac{2}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = \frac{2}{3}$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = \frac{\operatorname{atan}{\left(2 \right)}}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = \frac{\operatorname{atan}{\left(2 \right)}}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = \frac{2}{3}$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = \frac{\operatorname{atan}{\left(2 \right)}}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = \frac{\operatorname{atan}{\left(2 \right)}}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(2 x \right)}}{3 x}\right) = 0$$
More at x→-oo
The graph