Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of sin(9*x)*tan(6*x)/(1-cos(10*x))
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Limit of cot(5*x)*tan(3*x)
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Limit of log(sin(x)/x)
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Limit of (14-2*x)*log((11+5*x)/(13+5*x))
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Identical expressions
- sin(nine *x)*tan(six *x)/(one -cos(ten *x))
- sinus of (9 multiply by x) multiply by tangent of (6 multiply by x) divide by (1 minus co sinus of e of (10 multiply by x))
- sinus of (nine multiply by x) multiply by tangent of (six multiply by x) divide by (one minus co sinus of e of (ten multiply by x))
- sin(9x)tan(6x)/(1-cos(10x))
- sin9xtan6x/1-cos10x
- sin(9*x)*tan(6*x) divide by (1-cos(10*x))
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Similar expressions
- sin(9*x)*tan(6*x)/(1+cos(10*x))
Limit of the function sin(9*x)*tan(6*x)/(1-cos(10*x))
The solution
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[src]
/sin(9*x)*tan(6*x)\ lim |-----------------| x->0+\ 1 - cos(10*x) /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)} \tan{\left(6 x \right)}}{1 - \cos{\left(10 x \right)}}\right)$$
Limit((sin(9*x)*tan(6*x))/(1 - cos(10*x)), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(\sin{\left(9 x \right)} \tan{\left(6 x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(1 - \cos{\left(10 x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)} \tan{\left(6 x \right)}}{1 - \cos{\left(10 x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)} \tan{\left(6 x \right)}}{1 - \cos{\left(10 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(9 x \right)} \tan{\left(6 x \right)}}{\frac{d}{d x} \left(1 - \cos{\left(10 x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6 \sin{\left(9 x \right)} \tan^{2}{\left(6 x \right)} + 6 \sin{\left(9 x \right)} + 9 \cos{\left(9 x \right)} \tan{\left(6 x \right)}}{10 \sin{\left(10 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6 \sin{\left(9 x \right)} \tan^{2}{\left(6 x \right)} + 6 \sin{\left(9 x \right)} + 9 \cos{\left(9 x \right)} \tan{\left(6 x \right)}}{10 \sin{\left(10 x \right)}}\right)$$
=
$$\frac{27}{25}$$
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
$$\lim_{x \to 0^+}\left(\sin{\left(9 x \right)} \tan{\left(6 x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(1 - \cos{\left(10 x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)} \tan{\left(6 x \right)}}{1 - \cos{\left(10 x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)} \tan{\left(6 x \right)}}{1 - \cos{\left(10 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(9 x \right)} \tan{\left(6 x \right)}}{\frac{d}{d x} \left(1 - \cos{\left(10 x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6 \sin{\left(9 x \right)} \tan^{2}{\left(6 x \right)} + 6 \sin{\left(9 x \right)} + 9 \cos{\left(9 x \right)} \tan{\left(6 x \right)}}{10 \sin{\left(10 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6 \sin{\left(9 x \right)} \tan^{2}{\left(6 x \right)} + 6 \sin{\left(9 x \right)} + 9 \cos{\left(9 x \right)} \tan{\left(6 x \right)}}{10 \sin{\left(10 x \right)}}\right)$$
=
$$\frac{27}{25}$$
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]
/sin(9*x)*tan(6*x)\ lim |-----------------| x->0+\ 1 - cos(10*x) /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)} \tan{\left(6 x \right)}}{1 - \cos{\left(10 x \right)}}\right)$$
27 -- 25
$$\frac{27}{25}$$
= 1.08
/sin(9*x)*tan(6*x)\ lim |-----------------| x->0-\ 1 - cos(10*x) /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(9 x \right)} \tan{\left(6 x \right)}}{1 - \cos{\left(10 x \right)}}\right)$$
27 -- 25
$$\frac{27}{25}$$
= 1.08
= 1.08
The graph