Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-1+x)/(x+x^2)
-
Limit of log(-5+x)/log(e^x-e^5)
-
Limit of (-exp(-x)-2*x+exp(x))/(x-sin(x))
-
Limit of (e^(4*x)-e^(3*x))/(-sin(3*x)+sin(4*x))
- Integral of d{x}:
-
x/(1-cos(x))
-
Identical expressions
- x/(one -cos(x))
- x divide by (1 minus co sinus of e of (x))
- x divide by (one minus co sinus of e of (x))
- x/1-cosx
- x divide by (1-cos(x))
-
Similar expressions
- (1-cos(x)*cos(2*x)*cos(3*x))/(1-cos(x))
- (1-cos(x))/(1-cos(x/2))
- sin(x)*tan(x)/(1-cos(x))
- (-x*cos(x)+sin(x))/((1-cos(x))*sin(x))
- (-x+log(1+x))/(1-cos(x))
- x/(1+cos(x))
- x/(1-cosx)
Limit of the function x/(1-cos(x))
The solution
You have entered
[src]
/ x \ lim |----------| x->0+\1 - cos(x)/
$$\lim_{x \to 0^+}\left(\frac{x}{1 - \cos{\left(x \right)}}\right)$$
Limit(x/(1 - cos(x)), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} x = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(1 - \cos{\left(x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{x}{1 - \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \left(1 - \cos{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+} \frac{1}{\sin{\left(x \right)}}$$
=
$$\lim_{x \to 0^+} \frac{1}{\sin{\left(x \right)}}$$
=
$$\infty$$
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
$$\lim_{x \to 0^+} x = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(1 - \cos{\left(x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{x}{1 - \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \left(1 - \cos{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+} \frac{1}{\sin{\left(x \right)}}$$
=
$$\lim_{x \to 0^+} \frac{1}{\sin{\left(x \right)}}$$
=
$$\infty$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{x}{1 - \cos{\left(x \right)}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{1 - \cos{\left(x \right)}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{x}{1 - \cos{\left(x \right)}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{x}{1 - \cos{\left(x \right)}}\right) = - \frac{1}{-1 + \cos{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{1 - \cos{\left(x \right)}}\right) = - \frac{1}{-1 + \cos{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{1 - \cos{\left(x \right)}}\right) = -\infty$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{1 - \cos{\left(x \right)}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{x}{1 - \cos{\left(x \right)}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{x}{1 - \cos{\left(x \right)}}\right) = - \frac{1}{-1 + \cos{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{1 - \cos{\left(x \right)}}\right) = - \frac{1}{-1 + \cos{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{1 - \cos{\left(x \right)}}\right) = -\infty$$
More at x→-oo
One‐sided limits
[src]
/ x \ lim |----------| x->0+\1 - cos(x)/
$$\lim_{x \to 0^+}\left(\frac{x}{1 - \cos{\left(x \right)}}\right)$$
oo
$$\infty$$
= 302.00110375518
/ x \ lim |----------| x->0-\1 - cos(x)/
$$\lim_{x \to 0^-}\left(\frac{x}{1 - \cos{\left(x \right)}}\right)$$
-oo
$$-\infty$$
= -302.00110375518
= -302.00110375518
The graph