Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
- Limit of cot(5*pi*x)*log(x)
-
Limit of log(-1+x)/cot(pi*x)
-
Limit of sqrt(x)
- Limit of x^n
-
Identical expressions
- cot(five *pi*x)*log(x)
- cotangent of (5 multiply by Pi multiply by x) multiply by logarithm of (x)
- cotangent of (five multiply by Pi multiply by x) multiply by logarithm of (x)
- cot(5pix)log(x)
- cot5pixlogx
Limit of the function cot(5*pi*x)*log(x)
The solution
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lim (cot(5*pi*x)*log(x)) x->1+
$$\lim_{x \to 1^+}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right)$$
Limit(cot((5*pi)*x)*log(x), x, 1)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 1^+} \log{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+} \frac{1}{\cot{\left(5 \pi x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 1^+}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(5 \pi x \right)}}}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{\cot^{2}{\left(5 \pi x \right)}}{5 \pi x \left(- \cot^{2}{\left(5 \pi x \right)} - 1\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{\cot^{2}{\left(5 \pi x \right)}}{5 \pi x \left(- \cot^{2}{\left(5 \pi x \right)} - 1\right)}\right)$$
=
$$\frac{1}{5 \pi}$$
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
$$\lim_{x \to 1^+} \log{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+} \frac{1}{\cot{\left(5 \pi x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 1^+}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(5 \pi x \right)}}}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{\cot^{2}{\left(5 \pi x \right)}}{5 \pi x \left(- \cot^{2}{\left(5 \pi x \right)} - 1\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{\cot^{2}{\left(5 \pi x \right)}}{5 \pi x \left(- \cot^{2}{\left(5 \pi x \right)} - 1\right)}\right)$$
=
$$\frac{1}{5 \pi}$$
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)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 1^-}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right) = \frac{1}{5 \pi}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right) = \frac{1}{5 \pi}$$
$$\lim_{x \to \infty}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right)$$
More at x→-oo
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right) = \frac{1}{5 \pi}$$
$$\lim_{x \to \infty}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right)$$
More at x→-oo
One‐sided limits
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lim (cot(5*pi*x)*log(x)) x->1+
$$\lim_{x \to 1^+}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right)$$
1 ---- 5*pi
$$\frac{1}{5 \pi}$$
= 0.0636619772367581
lim (cot(5*pi*x)*log(x)) x->1-
$$\lim_{x \to 1^-}\left(\log{\left(x \right)} \cot{\left(5 \pi x \right)}\right)$$
1 ---- 5*pi
$$\frac{1}{5 \pi}$$
= 0.0636619772367581
= 0.0636619772367581