Mister Exam
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How to use it?
- Limit of the function:
- Limit of (1+2*x/3)^(3251035981008077*x/140737488355328)
-
Limit of (-8+x^2+2*x)/(-12+x+x^2)
-
Limit of (-1+sqrt(1-2*x+5*x^2)-x)/(3*x)
-
Limit of (-x+sin(x))/(x*sin(x))
-
Identical expressions
- (-x+sin(x))/(x*sin(x))
- ( minus x plus sinus of (x)) divide by (x multiply by sinus of (x))
- (-x+sin(x))/(xsin(x))
- -x+sinx/xsinx
- (-x+sin(x)) divide by (x*sin(x))
-
Similar expressions
- (x+sin(x))/(x*sin(x))
- (-x-sin(x))/(x*sin(x))
- (-x+sinx)/(x*sinx)
Limit of the function (-x+sin(x))/(x*sin(x))
The solution
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[src]
/-x + sin(x)\ lim |-----------| x->0+\ x*sin(x) /
$$\lim_{x \to 0^+}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right)$$
Limit((-x + sin(x))/((x*sin(x))), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(- x + \sin{\left(x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(x \sin{\left(x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x + \sin{\left(x \right)}\right)}{\frac{d}{d x} x \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)} - 1}{x \cos{\left(x \right)} + \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)} - 1}{x \cos{\left(x \right)} + \sin{\left(x \right)}}\right)$$
=
$$0$$
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
$$\lim_{x \to 0^+}\left(- x + \sin{\left(x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(x \sin{\left(x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x + \sin{\left(x \right)}\right)}{\frac{d}{d x} x \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)} - 1}{x \cos{\left(x \right)} + \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)} - 1}{x \cos{\left(x \right)} + \sin{\left(x \right)}}\right)$$
=
$$0$$
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]
/-x + sin(x)\ lim |-----------| x->0+\ x*sin(x) /
$$\lim_{x \to 0^+}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right)$$
0
$$0$$
= -2.31756905192471e-32
/-x + sin(x)\ lim |-----------| x->0-\ x*sin(x) /
$$\lim_{x \to 0^-}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right)$$
0
$$0$$
= 2.31756905192471e-32
= 2.31756905192471e-32
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right) = \frac{-1 + \sin{\left(1 \right)}}{\sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right) = \frac{-1 + \sin{\left(1 \right)}}{\sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right) = \frac{-1 + \sin{\left(1 \right)}}{\sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right) = \frac{-1 + \sin{\left(1 \right)}}{\sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- x + \sin{\left(x \right)}}{x \sin{\left(x \right)}}\right)$$
More at x→-oo
The graph