Mister Exam
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- Limit of the function:
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Limit of (-2+sqrt(x))/(-4+x^2)
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Limit of x+2*x^3+5*x^4-x^2/3
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Limit of (-x^3+2*x+5*x^4)/(1+x^4-8*x^3)
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Identical expressions
- tan(six *x)/sin(two *x)
- tangent of (6 multiply by x) divide by sinus of (2 multiply by x)
- tangent of (six multiply by x) divide by sinus of (two multiply by x)
- tan(6x)/sin(2x)
- tan6x/sin2x
- tan(6*x) divide by sin(2*x)
Limit of the function tan(6*x)/sin(2*x)
The solution
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[src]
/tan(6*x)\ lim |--------| x->0+\sin(2*x)/
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right)$$
Limit(tan(6*x)/sin(2*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \tan{\left(6 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(2 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(6 x \right)}}{\frac{d}{d x} \sin{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6 \tan^{2}{\left(6 x \right)} + 6}{2 \cos{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(3 \tan^{2}{\left(6 x \right)} + 3\right)$$
=
$$\lim_{x \to 0^+}\left(3 \tan^{2}{\left(6 x \right)} + 3\right)$$
=
$$3$$
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
$$\lim_{x \to 0^+} \tan{\left(6 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(2 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(6 x \right)}}{\frac{d}{d x} \sin{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{6 \tan^{2}{\left(6 x \right)} + 6}{2 \cos{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(3 \tan^{2}{\left(6 x \right)} + 3\right)$$
=
$$\lim_{x \to 0^+}\left(3 \tan^{2}{\left(6 x \right)} + 3\right)$$
=
$$3$$
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]
/tan(6*x)\ lim |--------| x->0+\sin(2*x)/
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right)$$
3
$$3$$
= 3.0
/tan(6*x)\ lim |--------| x->0-\sin(2*x)/
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right)$$
3
$$3$$
= 3.0
= 3.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right) = 3$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right) = 3$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right) = \frac{\tan{\left(6 \right)}}{\sin{\left(2 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right) = \frac{\tan{\left(6 \right)}}{\sin{\left(2 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right) = 3$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right) = \frac{\tan{\left(6 \right)}}{\sin{\left(2 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right) = \frac{\tan{\left(6 \right)}}{\sin{\left(2 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(6 x \right)}}{\sin{\left(2 x \right)}}\right)$$
More at x→-oo
The graph