Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of 6*x/tan(2*x)
-
Limit of 5*x*cot(2*x)
-
Limit of (5+2*n)/(3+2*n)
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Limit of -1+(-1+x)*(4+x)/x^2
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Identical expressions
- six *x/tan(two *x)
- 6 multiply by x divide by tangent of (2 multiply by x)
- six multiply by x divide by tangent of (two multiply by x)
- 6x/tan(2x)
- 6x/tan2x
- 6*x divide by tan(2*x)
-
Similar expressions
- x*e^(1-5*x)*sin(6*x)/tan(2*x)
- x*(-1+x*e^5)*sin(6*x)/tan(2*x)
- x*e^(-1+5*x)*sin(6*x)/tan(2*x)
- log(1-6*x)/tan(2*x)
- x*sin(6*x)/tan(2*x)^2
Limit of the function 6*x/tan(2*x)
The solution
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/ 6*x \ lim |--------| x->0+\tan(2*x)/
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
Limit((6*x)/tan(2*x), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
transform
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right) = \lim_{x \to 0^+}\left(\frac{6 x \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6 x}{\sin{\left(2 x \right)}}\right) \lim_{x \to 0^+} \cos{\left(2 x \right)} = \lim_{x \to 0^+}\left(\frac{6 x}{\sin{\left(2 x \right)}}\right)$$
Do replacement
$$u = 2 x$$
then
$$\lim_{x \to 0^+}\left(\frac{6 x}{\sin{\left(2 x \right)}}\right) = \lim_{u \to 0^+}\left(\frac{3 u}{\sin{\left(u \right)}}\right)$$
=
$$3 \lim_{u \to 0^+}\left(\frac{u}{\sin{\left(u \right)}}\right)$$
=
$$3 \left(\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)\right)^{-1}$$
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:
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right) = 3$$
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
transform
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right) = \lim_{x \to 0^+}\left(\frac{6 x \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6 x}{\sin{\left(2 x \right)}}\right) \lim_{x \to 0^+} \cos{\left(2 x \right)} = \lim_{x \to 0^+}\left(\frac{6 x}{\sin{\left(2 x \right)}}\right)$$
Do replacement
$$u = 2 x$$
then
$$\lim_{x \to 0^+}\left(\frac{6 x}{\sin{\left(2 x \right)}}\right) = \lim_{u \to 0^+}\left(\frac{3 u}{\sin{\left(u \right)}}\right)$$
=
$$3 \lim_{u \to 0^+}\left(\frac{u}{\sin{\left(u \right)}}\right)$$
=
$$3 \left(\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)\right)^{-1}$$
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:
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right) = 3$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(6 x\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \tan{\left(2 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 6 x}{\frac{d}{d x} \tan{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6}{2 \tan^{2}{\left(2 x \right)} + 2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6}{2 \tan^{2}{\left(2 x \right)} + 2}\right)$$
=
$$3$$
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
$$\lim_{x \to 0^+}\left(6 x\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \tan{\left(2 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 6 x}{\frac{d}{d x} \tan{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6}{2 \tan^{2}{\left(2 x \right)} + 2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6}{2 \tan^{2}{\left(2 x \right)} + 2}\right)$$
=
$$3$$
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]
/ 6*x \ lim |--------| x->0+\tan(2*x)/
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
3
$$3$$
= 3.0
/ 6*x \ lim |--------| x->0-\tan(2*x)/
$$\lim_{x \to 0^-}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
3
$$3$$
= 3.0
= 3.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right) = 3$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right) = 3$$
$$\lim_{x \to \infty}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right) = \frac{6}{\tan{\left(2 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right) = \frac{6}{\tan{\left(2 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right) = 3$$
$$\lim_{x \to \infty}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right) = \frac{6}{\tan{\left(2 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right) = \frac{6}{\tan{\left(2 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{6 x}{\tan{\left(2 x \right)}}\right)$$
More at x→-oo
The graph