Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of cos(x^2)
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Limit of (3+5*x)/(x^3-x-4*x^2)
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Limit of (5-2*x+5*x^2)/(-5+3*x^2)
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Limit of (5*x^2+15*x)/(3+x)
- Graphing y =:
- sqrt(x)/tan(x)
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Identical expressions
- sqrt(x)/tan(x)
- square root of (x) divide by tangent of (x)
- √(x)/tan(x)
- sqrtx/tanx
- sqrt(x) divide by tan(x)
Limit of the function sqrt(x)/tan(x)
The solution
You have entered
[src]
/ ___ \
|\/ x |
lim |------|
x->0+\tan(x)/
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
Limit(sqrt(x)/tan(x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sqrt{x} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \tan{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sqrt{x}}{\frac{d}{d x} \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{2 \sqrt{x} \left(\tan^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{2 \sqrt{x} \left(\tan^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\infty$$
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
$$\lim_{x \to 0^+} \sqrt{x} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \tan{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sqrt{x}}{\frac{d}{d x} \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{2 \sqrt{x} \left(\tan^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{2 \sqrt{x} \left(\tan^{2}{\left(x \right)} + 1\right)}\right)$$
=
$$\infty$$
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
$$\lim_{x \to 0^-}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right) = \frac{1}{\tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right) = \frac{1}{\tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right) = \frac{1}{\tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right) = \frac{1}{\tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
More at x→-oo
One‐sided limits
[src]
/ ___ \
|\/ x |
lim |------|
x->0+\tan(x)/
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
oo
$$\infty$$
= 12.2880260826679
/ ___ \
|\/ x |
lim |------|
x->0-\tan(x)/
$$\lim_{x \to 0^-}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
-oo*I
$$- \infty i$$
= (0.0 - 12.2880260826679j)
= (0.0 - 12.2880260826679j)
The graph