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e^(-1/x)*cot(x)
Limit of the function/ e^(-1/x)*cot(x)

Limit of the function e^(-1/x)*cot(x)

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     / -1        \
     | ---       |
     |  x        |
 lim \E   *cot(x)/
x->0+
lim⁡x→0+(e−1xcot⁡(x))\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \cot{\left(x \right)}\right) 
Limit(E^(-1/x)*cot(x), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+e−1x=0\lim_{x \to 0^+} e^{- \frac{1}{x}} = 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+(e−1xcot⁡(x))\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \cot{\left(x \right)}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(e−1xcot⁡(x))\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \cot{\left(x \right)}\right)
=
lim⁡x→0+(ddxe−1xddx1cot⁡(x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} e^{- \frac{1}{x}}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)
=
lim⁡x→0+(e−1xcot⁡2(x)x2(cot⁡2(x)+1))\lim_{x \to 0^+}\left(\frac{e^{- \frac{1}{x}} \cot^{2}{\left(x \right)}}{x^{2} \left(\cot^{2}{\left(x \right)} + 1\right)}\right)
=
lim⁡x→0+(e−1xcot⁡2(x)x2(cot⁡2(x)+1))\lim_{x \to 0^+}\left(\frac{e^{- \frac{1}{x}} \cot^{2}{\left(x \right)}}{x^{2} \left(\cot^{2}{\left(x \right)} + 1\right)}\right)
=
00
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
02468-8-6-4-2-1010-400000200000
Plot the graph
Rapid solution [src] 
0
00
Expand and simplify
One‐sided limits [src] 
     / -1        \
     | ---       |
     |  x        |
 lim \E   *cot(x)/
x->0+
lim⁡x→0+(e−1xcot⁡(x))\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \cot{\left(x \right)}\right) 
0
00 
= -1.92831438414116e-28
     / -1        \
     | ---       |
     |  x        |
 lim \E   *cot(x)/
x->0-
lim⁡x→0−(e−1xcot⁡(x))\lim_{x \to 0^-}\left(e^{- \frac{1}{x}} \cot{\left(x \right)}\right) 
-oo
−∞-\infty 
= 0.0258178018183778
= 0.0258178018183778
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(e−1xcot⁡(x))=0\lim_{x \to 0^-}\left(e^{- \frac{1}{x}} \cot{\left(x \right)}\right) = 0
More at x→0 from the left
lim⁡x→0+(e−1xcot⁡(x))=0\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \cot{\left(x \right)}\right) = 0
lim⁡x→∞(e−1xcot⁡(x))\lim_{x \to \infty}\left(e^{- \frac{1}{x}} \cot{\left(x \right)}\right)
More at x→oo
lim⁡x→1−(e−1xcot⁡(x))=1etan⁡(1)\lim_{x \to 1^-}\left(e^{- \frac{1}{x}} \cot{\left(x \right)}\right) = \frac{1}{e \tan{\left(1 \right)}}
More at x→1 from the left
lim⁡x→1+(e−1xcot⁡(x))=1etan⁡(1)\lim_{x \to 1^+}\left(e^{- \frac{1}{x}} \cot{\left(x \right)}\right) = \frac{1}{e \tan{\left(1 \right)}}
More at x→1 from the right
lim⁡x→−∞(e−1xcot⁡(x))\lim_{x \to -\infty}\left(e^{- \frac{1}{x}} \cot{\left(x \right)}\right)
More at x→-oo
Numerical answer [src] 
-1.92831438414116e-28
-1.92831438414116e-28
The graph
Limit of the function e^(-1/x)*cot(x)
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