Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of sin(3)^2/x
-
Limit of x^4+2*cos(x)^2
-
Limit of x*2^x*3^(-x)
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Limit of log(1+3*x^2)/(x^3-5*x^2)
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Identical expressions
- sin(x^ three)/x^ three
- sinus of (x cubed ) divide by x cubed
- sinus of (x to the power of three) divide by x to the power of three
- sin(x3)/x3
- sinx3/x3
- sin(x³)/x³
- sin(x to the power of 3)/x to the power of 3
- sinx^3/x^3
- sin(x^3) divide by x^3
Limit of the function sin(x^3)/x^3
The solution
You have entered
[src]
/ / 3\\
|sin\x /|
lim |-------|
x->0+| 3 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right)$$
Limit(sin(x^3)/(x^3), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(x^{3} \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x^{3} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x^{3} \right)}}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to 0^+} \cos{\left(x^{3} \right)}$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(x^{3} \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x^{3} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x^{3} \right)}}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to 0^+} \cos{\left(x^{3} \right)}$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = \sin{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = \sin{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = \sin{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = \sin{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/ / 3\\
|sin\x /|
lim |-------|
x->0+| 3 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right)$$
1
$$1$$
= 1
/ / 3\\
|sin\x /|
lim |-------|
x->0-| 3 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right)$$
1
$$1$$
= 1
= 1
The graph