Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of 6*x/tan(2*x)
-
Limit of 5*x*cot(2*x)
-
Limit of (5+2*n)/(3+2*n)
-
Limit of -1+(-1+x)*(4+x)/x^2
-
Identical expressions
- five *x*cot(two *x)
- 5 multiply by x multiply by cotangent of (2 multiply by x)
- five multiply by x multiply by cotangent of (two multiply by x)
- 5xcot(2x)
- 5xcot2x
Limit of the function 5*x*cot(2*x)
The solution
You have entered
[src]
lim (5*x*cot(2*x)) x->0+
$$\lim_{x \to 0^+}\left(5 x \cot{\left(2 x \right)}\right)$$
Limit((5*x)*cot(2*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(5 x\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(2 x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(5 x \cot{\left(2 x \right)}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(5 x \cot{\left(2 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 5 x}{\frac{d}{d x} \frac{1}{\cot{\left(2 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cot^{2}{\left(2 x \right)}}{2 \cot^{2}{\left(2 x \right)} + 2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cot^{2}{\left(2 x \right)}}{2 \cot^{2}{\left(2 x \right)} + 2}\right)$$
=
$$\frac{5}{2}$$
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
$$\lim_{x \to 0^+}\left(5 x\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(2 x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(5 x \cot{\left(2 x \right)}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(5 x \cot{\left(2 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 5 x}{\frac{d}{d x} \frac{1}{\cot{\left(2 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cot^{2}{\left(2 x \right)}}{2 \cot^{2}{\left(2 x \right)} + 2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cot^{2}{\left(2 x \right)}}{2 \cot^{2}{\left(2 x \right)} + 2}\right)$$
=
$$\frac{5}{2}$$
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]
lim (5*x*cot(2*x)) x->0+
$$\lim_{x \to 0^+}\left(5 x \cot{\left(2 x \right)}\right)$$
5/2
$$\frac{5}{2}$$
= 2.5
lim (5*x*cot(2*x)) x->0-
$$\lim_{x \to 0^-}\left(5 x \cot{\left(2 x \right)}\right)$$
5/2
$$\frac{5}{2}$$
= 2.5
= 2.5
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(5 x \cot{\left(2 x \right)}\right) = \frac{5}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(5 x \cot{\left(2 x \right)}\right) = \frac{5}{2}$$
$$\lim_{x \to \infty}\left(5 x \cot{\left(2 x \right)}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(5 x \cot{\left(2 x \right)}\right) = \frac{5}{\tan{\left(2 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(5 x \cot{\left(2 x \right)}\right) = \frac{5}{\tan{\left(2 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(5 x \cot{\left(2 x \right)}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(5 x \cot{\left(2 x \right)}\right) = \frac{5}{2}$$
$$\lim_{x \to \infty}\left(5 x \cot{\left(2 x \right)}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(5 x \cot{\left(2 x \right)}\right) = \frac{5}{\tan{\left(2 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(5 x \cot{\left(2 x \right)}\right) = \frac{5}{\tan{\left(2 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(5 x \cot{\left(2 x \right)}\right)$$
More at x→-oo
The graph