Mister Exam
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How to use it?
- Limit of the function:
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-
Limit of sqrt(x+sqrt(x+sqrt(x)))-sqrt(x)
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Limit of sin(x)/log(1+2*x)
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Limit of sin(pi*x/3)
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Identical expressions
- tan(m*x)/sin(n*x)
- tangent of (m multiply by x) divide by sinus of (n multiply by x)
- tan(mx)/sin(nx)
- tanmx/sinnx
- tan(m*x) divide by sin(n*x)
Limit of the function tan(m*x)/sin(n*x)
The solution
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/tan(m*x)\ lim |--------| x->0+\sin(n*x)/
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(m x \right)}}{\sin{\left(n x \right)}}\right)$$
Limit(tan(m*x)/sin(n*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \tan{\left(m x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(n x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(m x \right)}}{\sin{\left(n x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{\partial}{\partial x} \tan{\left(m x \right)}}{\frac{\partial}{\partial x} \sin{\left(n x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{m \left(\tan^{2}{\left(m x \right)} + 1\right)}{n \cos{\left(n x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{m}{n}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{m}{n}\right)$$
=
$$\frac{m}{n}$$
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
$$\lim_{x \to 0^+} \tan{\left(m x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(n x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(m x \right)}}{\sin{\left(n x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{\partial}{\partial x} \tan{\left(m x \right)}}{\frac{\partial}{\partial x} \sin{\left(n x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{m \left(\tan^{2}{\left(m x \right)} + 1\right)}{n \cos{\left(n x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{m}{n}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{m}{n}\right)$$
=
$$\frac{m}{n}$$
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)
One‐sided limits
[src]
/tan(m*x)\ lim |--------| x->0+\sin(n*x)/
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(m x \right)}}{\sin{\left(n x \right)}}\right)$$
m - n
$$\frac{m}{n}$$
/tan(m*x)\ lim |--------| x->0-\sin(n*x)/
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(m x \right)}}{\sin{\left(n x \right)}}\right)$$
m - n
$$\frac{m}{n}$$
m/n