Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (5+x-3*x^2)/(4-x+2*x^2)
-
Limit of ((5+2*x)/(-1+2*x))^(3-x)
-
Limit of (20+x^2-9*x)/(-4+x^2-3*x)
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Limit of (4+x^2+4*x)/(-4+x^2)
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Identical expressions
- csc(x)*sin(sin(x))
- csc(x) multiply by sinus of ( sinus of (x))
- csc(x)sin(sin(x))
- cscxsinsinx
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Similar expressions
- cosec(x)*sin(sin(x))
- cosecx*sin(sinx)
Limit of the function csc(x)*sin(sin(x))
The solution
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lim (csc(x)*sin(sin(x))) x->0+
$$\lim_{x \to 0^+}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right)$$
Limit(csc(x)*sin(sin(x)), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(\sin{\left(x \right)} \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\csc{\left(x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(\sin{\left(x \right)} \right)}}{\frac{d}{d x} \frac{1}{\csc{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}}{\cot{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\csc{\left(x \right)}}{\cot{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\csc{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right) \csc{\left(x \right)}}{\cot^{3}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right) \csc{\left(x \right)}}{\cot^{3}{\left(x \right)}}\right)$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(\sin{\left(x \right)} \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\csc{\left(x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(\sin{\left(x \right)} \right)}}{\frac{d}{d x} \frac{1}{\csc{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}}{\cot{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\csc{\left(x \right)}}{\cot{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\csc{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right) \csc{\left(x \right)}}{\cot^{3}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right) \csc{\left(x \right)}}{\cot^{3}{\left(x \right)}}\right)$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
The graph
One‐sided limits
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lim (csc(x)*sin(sin(x))) x->0+
$$\lim_{x \to 0^+}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right)$$
1
$$1$$
= 1.0
lim (csc(x)*sin(sin(x))) x->0-
$$\lim_{x \to 0^-}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right)$$
1
$$1$$
= 1.0
= 1.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right) = 1$$
$$\lim_{x \to \infty}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right) = \frac{\sin{\left(\sin{\left(1 \right)} \right)}}{\sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right) = \frac{\sin{\left(\sin{\left(1 \right)} \right)}}{\sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right) = 1$$
$$\lim_{x \to \infty}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right) = \frac{\sin{\left(\sin{\left(1 \right)} \right)}}{\sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right) = \frac{\sin{\left(\sin{\left(1 \right)} \right)}}{\sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\sin{\left(\sin{\left(x \right)} \right)} \csc{\left(x \right)}\right)$$
More at x→-oo
The graph