Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
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Limit of (-1+e^x)/sin(x)
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Limit of ((-1+x)/(4+x))^(2+3*x)
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Limit of (1+n)/(2+n)
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Limit of (3+2*x)/(1+5*x)
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Identical expressions
- (- two +sqrt(four +x))/(three *atan(x))
- ( minus 2 plus square root of (4 plus x)) divide by (3 multiply by arc tangent of gent of (x))
- ( minus two plus square root of (four plus x)) divide by (three multiply by arc tangent of gent of (x))
- (-2+√(4+x))/(3*atan(x))
- (-2+sqrt(4+x))/(3atan(x))
- -2+sqrt4+x/3atanx
- (-2+sqrt(4+x)) divide by (3*atan(x))
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Similar expressions
- (-2+sqrt(4-x))/(3*atan(x))
- (-2-sqrt(4+x))/(3*atan(x))
- (2+sqrt(4+x))/(3*atan(x))
- (-2+sqrt(4+x))/(3*arctan(x))
- (-2+sqrt(4+x))/(3*arctanx)
Limit of the function (-2+sqrt(4+x))/(3*atan(x))
The solution
You have entered
[src]
/ _______\
|-2 + \/ 4 + x |
lim |--------------|
x->0+\ 3*atan(x) /
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right)$$
Limit((-2 + sqrt(4 + x))/((3*atan(x))), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(\sqrt{x + 4} - 2\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(3 \operatorname{atan}{\left(x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sqrt{x + 4} - 2\right)}{\frac{d}{d x} 3 \operatorname{atan}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{x^{2}}{3} + \frac{1}{3}}{2 \sqrt{x + 4}}\right)$$
=
$$\lim_{x \to 0^+} \frac{1}{12}$$
=
$$\lim_{x \to 0^+} \frac{1}{12}$$
=
$$\frac{1}{12}$$
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
$$\lim_{x \to 0^+}\left(\sqrt{x + 4} - 2\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(3 \operatorname{atan}{\left(x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sqrt{x + 4} - 2\right)}{\frac{d}{d x} 3 \operatorname{atan}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{x^{2}}{3} + \frac{1}{3}}{2 \sqrt{x + 4}}\right)$$
=
$$\lim_{x \to 0^+} \frac{1}{12}$$
=
$$\lim_{x \to 0^+} \frac{1}{12}$$
=
$$\frac{1}{12}$$
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 + \/ 4 + x |
lim |--------------|
x->0+\ 3*atan(x) /
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right)$$
1/12
$$\frac{1}{12}$$
= 0.0833333333333333
/ _______\
|-2 + \/ 4 + x |
lim |--------------|
x->0-\ 3*atan(x) /
$$\lim_{x \to 0^-}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right)$$
1/12
$$\frac{1}{12}$$
= 0.0833333333333333
= 0.0833333333333333
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right) = \frac{1}{12}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right) = \frac{1}{12}$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right) = \frac{-8 + 4 \sqrt{5}}{3 \pi}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right) = \frac{-8 + 4 \sqrt{5}}{3 \pi}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right) = - \infty i$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right) = \frac{1}{12}$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right) = \frac{-8 + 4 \sqrt{5}}{3 \pi}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right) = \frac{-8 + 4 \sqrt{5}}{3 \pi}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x + 4} - 2}{3 \operatorname{atan}{\left(x \right)}}\right) = - \infty i$$
More at x→-oo
The graph