Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of 1-x-sin(x/2)/pi
-
Limit of (1-cos(x)^2)/x
-
Limit of (1-cos(4*x))/(4*x^2)
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Limit of (1+8*x)^(1/x)
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Identical expressions
- (one -cos(x)^ two)/x
- (1 minus co sinus of e of (x) squared ) divide by x
- (one minus co sinus of e of (x) to the power of two) divide by x
- (1-cos(x)2)/x
- 1-cosx2/x
- (1-cos(x)²)/x
- (1-cos(x) to the power of 2)/x
- 1-cosx^2/x
- (1-cos(x)^2) divide by x
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Similar expressions
- (1+cos(x)^2)/x
- (1-cos(x))^2/x^4
- (1-cos(x^2))/x^2
- (1-cosx^2)/x
Limit of the function (1-cos(x)^2)/x
The solution
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[src]
/ 2 \
|1 - cos (x)|
lim |-----------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right)$$
Limit((1 - cos(x)^2)/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(1 - \cos^{2}{\left(x \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \cos^{2}{\left(x \right)}\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(2 \sin{\left(x \right)} \cos{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(2 \sin{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(2 \sin{\left(x \right)}\right)$$
=
$$0$$
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
$$\lim_{x \to 0^+}\left(1 - \cos^{2}{\left(x \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \cos^{2}{\left(x \right)}\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(2 \sin{\left(x \right)} \cos{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(2 \sin{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(2 \sin{\left(x \right)}\right)$$
=
$$0$$
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]
/ 2 \
|1 - cos (x)|
lim |-----------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right)$$
0
$$0$$
= -7.34778436425346e-33
/ 2 \
|1 - cos (x)|
lim |-----------|
x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right)$$
0
$$0$$
= 7.34778436425346e-33
= 7.34778436425346e-33
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right) = 1 - \cos^{2}{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right) = 1 - \cos^{2}{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right) = 1 - \cos^{2}{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right) = 1 - \cos^{2}{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{1 - \cos^{2}{\left(x \right)}}{x}\right) = 0$$
More at x→-oo
The graph