Mister Exam
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How to use it?
- Limit of the function:
-
Limit of (-5+7*n)/(2+n)
-
Limit of (-64+4^x)/(-3+x)
-
Limit of (-7+3*x^2+5*x)/(1+x+3*x^2)
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Limit of (-2+2*x^3+7*x)/(-4-x+3*x^3)
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Identical expressions
- sin(sin(sin(x)))/x
- sinus of ( sinus of ( sinus of (x))) divide by x
- sinsinsinx/x
- sin(sin(sin(x))) divide by x
-
Similar expressions
- sin(sin(sinx))/x
Limit of the function sin(sin(sin(x)))/x
The solution
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[src]
/sin(sin(sin(x)))\ lim |----------------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right)$$
Limit(sin(sin(sin(x)))/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} \cos{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}\right)$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$1$$
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
$$\lim_{x \to 0^+} \sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} \cos{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}\right)$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$1$$
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(sin(x)))\ lim |----------------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right)$$
1
$$1$$
= 1.0
/sin(sin(sin(x)))\ lim |----------------| x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right)$$
1
$$1$$
= 1.0
= 1.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = 0$$
More at x→-oo
The graph