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

Limit of the function cos(1/x)

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The solution

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        /  1\
 lim cos|1*-|
x->0+   \  x/
lim⁡x→0+cos⁡(1⋅1x)\lim_{x \to 0^+} \cos{\left(1 \cdot \frac{1}{x} \right)} 
Limit(cos(1/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
02468-8-6-4-2-10102-2
Plot the graph
Rapid solution [src] 
<-1, 1>
⟨−1,1⟩\left\langle -1, 1\right\rangle
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−cos⁡(1⋅1x)=⟨−1,1⟩\lim_{x \to 0^-} \cos{\left(1 \cdot \frac{1}{x} \right)} = \left\langle -1, 1\right\rangle
More at x→0 from the left
lim⁡x→0+cos⁡(1⋅1x)=⟨−1,1⟩\lim_{x \to 0^+} \cos{\left(1 \cdot \frac{1}{x} \right)} = \left\langle -1, 1\right\rangle
lim⁡x→∞cos⁡(1⋅1x)=1\lim_{x \to \infty} \cos{\left(1 \cdot \frac{1}{x} \right)} = 1
More at x→oo
lim⁡x→1−cos⁡(1⋅1x)=cos⁡(1)\lim_{x \to 1^-} \cos{\left(1 \cdot \frac{1}{x} \right)} = \cos{\left(1 \right)}
More at x→1 from the left
lim⁡x→1+cos⁡(1⋅1x)=cos⁡(1)\lim_{x \to 1^+} \cos{\left(1 \cdot \frac{1}{x} \right)} = \cos{\left(1 \right)}
More at x→1 from the right
lim⁡x→−∞cos⁡(1⋅1x)=1\lim_{x \to -\infty} \cos{\left(1 \cdot \frac{1}{x} \right)} = 1
More at x→-oo
One‐sided limits [src] 
        /  1\
 lim cos|1*-|
x->0+   \  x/
lim⁡x→0+cos⁡(1⋅1x)\lim_{x \to 0^+} \cos{\left(1 \cdot \frac{1}{x} \right)} 
<-1, 1>
⟨−1,1⟩\left\langle -1, 1\right\rangle 
= -1.83036708622755e-76
        /  1\
 lim cos|1*-|
x->0-   \  x/
lim⁡x→0−cos⁡(1⋅1x)\lim_{x \to 0^-} \cos{\left(1 \cdot \frac{1}{x} \right)} 
<-1, 1>
⟨−1,1⟩\left\langle -1, 1\right\rangle 
= -1.83036708622755e-76
= -1.83036708622755e-76
Numerical answer [src] 
-1.83036708622755e-76
-1.83036708622755e-76
The graph
Limit of the function cos(1/x)
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