Mister Exam

Other calculators:

cot(x)*log(x+e^x)
Limit of the function/ cot(x)*log(x+e^x)

Limit of the function cot(x)*log(x+e^x)

at →
v

For end points:

The graph:

from to

Piecewise:

The solution

You have entered [src] 
     /          /     x\\
 lim \cot(x)*log\x + E //
x->0+
lim⁡x→0+(log⁡(ex+x)cot⁡(x))\lim_{x \to 0^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) 
Limit(cot(x)*log(x + E^x), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+log⁡(x+ex)=0\lim_{x \to 0^+} \log{\left(x + e^{x} \right)} = 0
and limit for the denominator is
lim⁡x→0+1cot⁡(x)=0\lim_{x \to 0^+} \frac{1}{\cot{\left(x \right)}} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(log⁡(ex+x)cot⁡(x))\lim_{x \to 0^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(log⁡(x+ex)cot⁡(x))\lim_{x \to 0^+}\left(\log{\left(x + e^{x} \right)} \cot{\left(x \right)}\right)
=
lim⁡x→0+(ddxlog⁡(x+ex)ddx1cot⁡(x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(x + e^{x} \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)
=
lim⁡x→0+((ex+1)cot⁡2(x)(x+ex)(cot⁡2(x)+1))\lim_{x \to 0^+}\left(\frac{\left(e^{x} + 1\right) \cot^{2}{\left(x \right)}}{\left(x + e^{x}\right) \left(\cot^{2}{\left(x \right)} + 1\right)}\right)
=
lim⁡x→0+(2cot⁡2(x)cot⁡2(x)+1)\lim_{x \to 0^+}\left(\frac{2 \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)
=
lim⁡x→0+(ddx1cot⁡2(x)+1ddx12cot⁡2(x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)}}}\right)
=
lim⁡x→0+(1(2cot⁡2(x)+2)(−cot⁡4(x)−4cot⁡6(x)−4cot⁡4(x)−2cot⁡2(x)−4cot⁡6(x)−4cot⁡4(x)−1−4cot⁡6(x)−4cot⁡4(x)))\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{\cot^{4}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{1}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)
=
lim⁡x→0+(ddx12cot⁡2(x)+2ddx(−cot⁡4(x)−4cot⁡6(x)−4cot⁡4(x)−2cot⁡2(x)−4cot⁡6(x)−4cot⁡4(x)−1−4cot⁡6(x)−4cot⁡4(x)))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{\cot^{4}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{1}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)
=
lim⁡x→0+(−2(−2cot⁡2(x)−2)cot⁡(x)(2cot⁡2(x)+2)2(−(4(−6cot⁡2(x)−6)cot⁡5(x)+4(−4cot⁡2(x)−4)cot⁡3(x))cot⁡4(x)(−4cot⁡6(x)−4cot⁡4(x))2−2(4(−6cot⁡2(x)−6)cot⁡5(x)+4(−4cot⁡2(x)−4)cot⁡3(x))cot⁡2(x)(−4cot⁡6(x)−4cot⁡4(x))2−4(−6cot⁡2(x)−6)cot⁡5(x)+4(−4cot⁡2(x)−4)cot⁡3(x)(−4cot⁡6(x)−4cot⁡4(x))2−(−4cot⁡2(x)−4)cot⁡3(x)−4cot⁡6(x)−4cot⁡4(x)−2(−2cot⁡2(x)−2)cot⁡(x)−4cot⁡6(x)−4cot⁡4(x)))\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)
=
lim⁡x→0+(−2(−2cot⁡2(x)−2)cot⁡(x)(2cot⁡2(x)+2)2(−(4(−6cot⁡2(x)−6)cot⁡5(x)+4(−4cot⁡2(x)−4)cot⁡3(x))cot⁡4(x)(−4cot⁡6(x)−4cot⁡4(x))2−2(4(−6cot⁡2(x)−6)cot⁡5(x)+4(−4cot⁡2(x)−4)cot⁡3(x))cot⁡2(x)(−4cot⁡6(x)−4cot⁡4(x))2−4(−6cot⁡2(x)−6)cot⁡5(x)+4(−4cot⁡2(x)−4)cot⁡3(x)(−4cot⁡6(x)−4cot⁡4(x))2−(−4cot⁡2(x)−4)cot⁡3(x)−4cot⁡6(x)−4cot⁡4(x)−2(−2cot⁡2(x)−2)cot⁡(x)−4cot⁡6(x)−4cot⁡4(x)))\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)
=
22
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
The graph
02468-8-6-4-2-1010-500500
Plot the graph
Rapid solution [src] 
2
22
Expand and simplify
One‐sided limits [src] 
     /          /     x\\
 lim \cot(x)*log\x + E //
x->0+
lim⁡x→0+(log⁡(ex+x)cot⁡(x))\lim_{x \to 0^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) 
2
22 
= 2.0
     /          /     x\\
 lim \cot(x)*log\x + E //
x->0-
lim⁡x→0−(log⁡(ex+x)cot⁡(x))\lim_{x \to 0^-}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) 
2
22 
= 2.0
= 2.0
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(log⁡(ex+x)cot⁡(x))=2\lim_{x \to 0^-}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) = 2
More at x→0 from the left
lim⁡x→0+(log⁡(ex+x)cot⁡(x))=2\lim_{x \to 0^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) = 2
lim⁡x→∞(log⁡(ex+x)cot⁡(x))\lim_{x \to \infty}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right)
More at x→oo
lim⁡x→1−(log⁡(ex+x)cot⁡(x))=log⁡(1+e)tan⁡(1)\lim_{x \to 1^-}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) = \frac{\log{\left(1 + e \right)}}{\tan{\left(1 \right)}}
More at x→1 from the left
lim⁡x→1+(log⁡(ex+x)cot⁡(x))=log⁡(1+e)tan⁡(1)\lim_{x \to 1^+}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right) = \frac{\log{\left(1 + e \right)}}{\tan{\left(1 \right)}}
More at x→1 from the right
lim⁡x→−∞(log⁡(ex+x)cot⁡(x))\lim_{x \to -\infty}\left(\log{\left(e^{x} + x \right)} \cot{\left(x \right)}\right)
More at x→-oo
Numerical answer [src] 
2.0
2.0
The graph
Limit of the function cot(x)*log(x+e^x)
    © Mister Exam – Calculator