Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (x-sin(x))/(x-tan(x))
-
Limit of tanh(n)^(1/n)
-
Limit of csch(x)
-
Limit of sqrt(2)/2
- Graphing y =:
- (x-sin(x))/(x-tan(x))
-
Identical expressions
- (x-sin(x))/(x-tan(x))
- (x minus sinus of (x)) divide by (x minus tangent of (x))
- x-sinx/x-tanx
- (x-sin(x)) divide by (x-tan(x))
-
Similar expressions
- (x+sin(x))/(x-tan(x))
- (x-sin(x))/(x+tan(x))
- x-sin(x)/(x-tan(x))
- (x-sinx)/(x-tan(x))
Limit of the function (x-sin(x))/(x-tan(x))
The solution
You have entered
[src]
/x - sin(x)\ lim |----------| x->-oo\x - tan(x)/
$$\lim_{x \to -\infty}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right)$$
Limit((x - sin(x))/(x - tan(x)), x, -oo)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to -\infty}\left(x - \sin{\left(x \right)}\right) = -\infty$$
and limit for the denominator is
$$\lim_{x \to -\infty}\left(x - \tan{\left(x \right)}\right) = -\infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to -\infty}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} \left(x - \sin{\left(x \right)}\right)}{\frac{d}{d x} \left(x - \tan{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to -\infty}\left(- \frac{1 - \cos{\left(x \right)}}{\tan^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to -\infty}\left(\left(1 - \cos{\left(x \right)}\right) \left\langle -\infty, 0\right\rangle\right)$$
=
$$\lim_{x \to -\infty}\left(\left(1 - \cos{\left(x \right)}\right) \left\langle -\infty, 0\right\rangle\right)$$
=
$$1$$
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)
-oo/-oo,
i.e. limit for the numerator is
$$\lim_{x \to -\infty}\left(x - \sin{\left(x \right)}\right) = -\infty$$
and limit for the denominator is
$$\lim_{x \to -\infty}\left(x - \tan{\left(x \right)}\right) = -\infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to -\infty}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} \left(x - \sin{\left(x \right)}\right)}{\frac{d}{d x} \left(x - \tan{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to -\infty}\left(- \frac{1 - \cos{\left(x \right)}}{\tan^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to -\infty}\left(\left(1 - \cos{\left(x \right)}\right) \left\langle -\infty, 0\right\rangle\right)$$
=
$$\lim_{x \to -\infty}\left(\left(1 - \cos{\left(x \right)}\right) \left\langle -\infty, 0\right\rangle\right)$$
=
$$1$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to -\infty}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right) = 1$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right) = - \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right) = - \frac{1}{2}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right) = \frac{1 - \sin{\left(1 \right)}}{1 - \tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right) = \frac{1 - \sin{\left(1 \right)}}{1 - \tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to \infty}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right) = 1$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right) = - \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right) = - \frac{1}{2}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right) = \frac{1 - \sin{\left(1 \right)}}{1 - \tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - \sin{\left(x \right)}}{x - \tan{\left(x \right)}}\right) = \frac{1 - \sin{\left(1 \right)}}{1 - \tan{\left(1 \right)}}$$
More at x→1 from the right
The graph