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cos(1/x)/x
Limit of the function/ cos(1/x)/x

Limit of the function cos(1/x)/x

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The solution

You have entered [src] 
     /   /1\\
     |cos|-||
     |   \x/|
 lim |------|
x->0+\  x   /
lim⁡x→0+(cos⁡(1x)x)\lim_{x \to 0^+}\left(\frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) 
Limit(cos(1/x)/x, x, 0)
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
02468-8-6-4-2-1010-2020
Plot the graph
Rapid solution [src] 
<-oo, oo>
⟨−∞,∞⟩\left\langle -\infty, \infty\right\rangle
Expand and simplify
One‐sided limits [src] 
     /   /1\\
     |cos|-||
     |   \x/|
 lim |------|
x->0+\  x   /
lim⁡x→0+(cos⁡(1x)x)\lim_{x \to 0^+}\left(\frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) 
<-oo, oo>
⟨−∞,∞⟩\left\langle -\infty, \infty\right\rangle 
= -1.99142122683281e-74
     /   /1\\
     |cos|-||
     |   \x/|
 lim |------|
x->0-\  x   /
lim⁡x→0−(cos⁡(1x)x)\lim_{x \to 0^-}\left(\frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) 
<-oo, oo>
⟨−∞,∞⟩\left\langle -\infty, \infty\right\rangle 
= 1.99142122683281e-74
= 1.99142122683281e-74
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(cos⁡(1x)x)=⟨−∞,∞⟩\lim_{x \to 0^-}\left(\frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) = \left\langle -\infty, \infty\right\rangle
More at x→0 from the left
lim⁡x→0+(cos⁡(1x)x)=⟨−∞,∞⟩\lim_{x \to 0^+}\left(\frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(cos⁡(1x)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) = 0
More at x→oo
lim⁡x→1−(cos⁡(1x)x)=cos⁡(1)\lim_{x \to 1^-}\left(\frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) = \cos{\left(1 \right)}
More at x→1 from the left
lim⁡x→1+(cos⁡(1x)x)=cos⁡(1)\lim_{x \to 1^+}\left(\frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) = \cos{\left(1 \right)}
More at x→1 from the right
lim⁡x→−∞(cos⁡(1x)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) = 0
More at x→-oo
Numerical answer [src] 
-1.99142122683281e-74
-1.99142122683281e-74
The graph
Limit of the function cos(1/x)/x
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