Mister Exam
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- Limit of the function:
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Limit of tan(15*x)/sin(3*x)
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Limit of sin(3*x)/sin(x)
-
Limit of n/factorial(n)
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Limit of tan(3*x)
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Identical expressions
- tan(fifteen *x)/sin(three *x)
- tangent of (15 multiply by x) divide by sinus of (3 multiply by x)
- tangent of (fifteen multiply by x) divide by sinus of (three multiply by x)
- tan(15x)/sin(3x)
- tan15x/sin3x
- tan(15*x) divide by sin(3*x)
Limit of the function tan(15*x)/sin(3*x)
The solution
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[src]
/tan(15*x)\ lim |---------| x->0+\ sin(3*x)/
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(15 x \right)}}{\sin{\left(3 x \right)}}\right)$$
Limit(tan(15*x)/sin(3*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \tan{\left(15 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(3 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(15 x \right)}}{\sin{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(15 x \right)}}{\frac{d}{d x} \sin{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{15 \tan^{2}{\left(15 x \right)} + 15}{3 \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(5 \tan^{2}{\left(15 x \right)} + 5\right)$$
=
$$\lim_{x \to 0^+}\left(5 \tan^{2}{\left(15 x \right)} + 5\right)$$
=
$$5$$
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
$$\lim_{x \to 0^+} \tan{\left(15 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(3 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(15 x \right)}}{\sin{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(15 x \right)}}{\frac{d}{d x} \sin{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{15 \tan^{2}{\left(15 x \right)} + 15}{3 \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(5 \tan^{2}{\left(15 x \right)} + 5\right)$$
=
$$\lim_{x \to 0^+}\left(5 \tan^{2}{\left(15 x \right)} + 5\right)$$
=
$$5$$
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]
/tan(15*x)\ lim |---------| x->0+\ sin(3*x)/
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(15 x \right)}}{\sin{\left(3 x \right)}}\right)$$
5
$$5$$
= 5.0
/tan(15*x)\ lim |---------| x->0-\ sin(3*x)/
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(15 x \right)}}{\sin{\left(3 x \right)}}\right)$$
5
$$5$$
= 5.0
= 5.0
The graph