Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-cos(5*x)+cos(3*x))/x^2
-
Limit of 2-4*sqrt(x)-sqrt(2)/2
-
Limit of (-12+x^2-4*x)/(48+x^2-14*x)
-
Limit of (2+x^2+3*x)/(3+x^2+4*x)
-
Identical expressions
- (one -cos(x))/(two *x)
- (1 minus co sinus of e of (x)) divide by (2 multiply by x)
- (one minus co sinus of e of (x)) divide by (two multiply by x)
- (1-cos(x))/(2x)
- 1-cosx/2x
- (1-cos(x)) divide by (2*x)
-
Similar expressions
- (1-cos(x))/(2*x*tan(2*x))
- (1-cos(x))/(2*x^2)
- (1+cos(x))/(2*x)
- (1-cosx)/(2*x)
Limit of the function (1-cos(x))/(2*x)
The solution
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[src]
/1 - cos(x)\ lim |----------| x->0+\ 2*x /
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right)$$
Limit((1 - cos(x))/((2*x)), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(1 - \cos{\left(x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(2 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(x \right)}\right)}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{2}\right)$$
=
$$0$$
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
$$\lim_{x \to 0^+}\left(1 - \cos{\left(x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(2 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(x \right)}\right)}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{2}\right)$$
=
$$0$$
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]
/1 - cos(x)\ lim |----------| x->0+\ 2*x /
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right)$$
0
$$0$$
= -5.86873299214896e-33
/1 - cos(x)\ lim |----------| x->0-\ 2*x /
$$\lim_{x \to 0^-}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right)$$
0
$$0$$
= 5.86873299214896e-33
= 5.86873299214896e-33
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right) = \frac{1}{2} - \frac{\cos{\left(1 \right)}}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right) = \frac{1}{2} - \frac{\cos{\left(1 \right)}}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right) = \frac{1}{2} - \frac{\cos{\left(1 \right)}}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right) = \frac{1}{2} - \frac{\cos{\left(1 \right)}}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{1 - \cos{\left(x \right)}}{2 x}\right) = 0$$
More at x→-oo
The graph