Mister Exam
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How to use it?
- Limit of the function:
-
Limit of ((-4+3*x)/(2+3*x))^(1+x)/3
-
Limit of (-16+2^x)/(-1+5*sqrt(x)*(5-x))
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Limit of (-14+x^2-5*x)/(-6+x+2*x^2)
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Limit of (3+x^2+4*x)/(1+x^3)
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Identical expressions
- sin(sin(x))/sin(x)
- sinus of ( sinus of (x)) divide by sinus of (x)
- sinsinx/sinx
- sin(sin(x)) divide by sin(x)
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Similar expressions
- sin(sinx)/sinx
Limit of the function sin(sin(x))/sin(x)
The solution
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[src]
/sin(sin(x))\ lim |-----------| x->0+\ sin(x) /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right)$$
Limit(sin(sin(x))/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
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(\sin{\left(x \right)} \right)}}{\frac{d}{d x} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+} \cos{\left(\sin{\left(x \right)} \right)}$$
=
$$\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}$$
=
$$\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}$$
=
$$1$$
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
$$\lim_{x \to 0^+} \sin{\left(\sin{\left(x \right)} \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(\sin{\left(x \right)} \right)}}{\frac{d}{d x} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+} \cos{\left(\sin{\left(x \right)} \right)}$$
=
$$\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}$$
=
$$\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}$$
=
$$1$$
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]
/sin(sin(x))\ lim |-----------| x->0+\ sin(x) /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right)$$
1
$$1$$
= 1.0
/sin(sin(x))\ lim |-----------| x->0-\ sin(x) /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right)$$
1
$$1$$
= 1.0
= 1.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) = \frac{\sin{\left(\sin{\left(1 \right)} \right)}}{\sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) = \frac{\sin{\left(\sin{\left(1 \right)} \right)}}{\sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) = \frac{\sin{\left(\sin{\left(1 \right)} \right)}}{\sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) = \frac{\sin{\left(\sin{\left(1 \right)} \right)}}{\sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right)$$
More at x→-oo
The graph