Mister Exam
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How to use it?
- Limit of the function:
-
Limit of (2*x/(-3+2*x))^(3*x)
-
Limit of (1-cos(8*x))/x^2
-
Limit of (-sin(x)+tan(x))/sin(x)^3
-
Limit of (-asin(x)+2*x)/(2*x+atan(x))
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Identical expressions
- tan(eight *x)/sin(five *x)
- tangent of (8 multiply by x) divide by sinus of (5 multiply by x)
- tangent of (eight multiply by x) divide by sinus of (five multiply by x)
- tan(8x)/sin(5x)
- tan8x/sin5x
- tan(8*x) divide by sin(5*x)
-
Similar expressions
- x*tan(8*x)/(sin(5*x)+sin(6*x))
Limit of the function tan(8*x)/sin(5*x)
The solution
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[src]
/tan(8*x)\ lim |--------| x->0+\sin(5*x)/
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right)$$
Limit(tan(8*x)/sin(5*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \tan{\left(8 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(5 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(8 x \right)}}{\frac{d}{d x} \sin{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{8 \tan^{2}{\left(8 x \right)} + 8}{5 \cos{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{8 \tan^{2}{\left(8 x \right)}}{5} + \frac{8}{5}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{8 \tan^{2}{\left(8 x \right)}}{5} + \frac{8}{5}\right)$$
=
$$\frac{8}{5}$$
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
$$\lim_{x \to 0^+} \tan{\left(8 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(5 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(8 x \right)}}{\frac{d}{d x} \sin{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{8 \tan^{2}{\left(8 x \right)} + 8}{5 \cos{\left(5 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{8 \tan^{2}{\left(8 x \right)}}{5} + \frac{8}{5}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{8 \tan^{2}{\left(8 x \right)}}{5} + \frac{8}{5}\right)$$
=
$$\frac{8}{5}$$
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(8*x)\ lim |--------| x->0+\sin(5*x)/
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right)$$
8/5
$$\frac{8}{5}$$
= 1.6
/tan(8*x)\ lim |--------| x->0-\sin(5*x)/
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right)$$
8/5
$$\frac{8}{5}$$
= 1.6
= 1.6
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) = \frac{8}{5}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) = \frac{8}{5}$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) = \frac{\tan{\left(8 \right)}}{\sin{\left(5 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) = \frac{\tan{\left(8 \right)}}{\sin{\left(5 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) = \frac{8}{5}$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) = \frac{\tan{\left(8 \right)}}{\sin{\left(5 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) = \frac{\tan{\left(8 \right)}}{\sin{\left(5 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right)$$
More at x→-oo
The graph