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cos(pi*x)
Limit of the function/ cos(pi*x)

Limit of the function cos(pi*x)

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The solution

You have entered [src] 
 lim cos(pi*x)
x->5+
lim⁡x→5+cos⁡(πx)\lim_{x \to 5^+} \cos{\left(\pi x \right)} 
Limit(cos(pi*x), x, 5)
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
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→5−cos⁡(πx)=−1\lim_{x \to 5^-} \cos{\left(\pi x \right)} = -1
More at x→5 from the left
lim⁡x→5+cos⁡(πx)=−1\lim_{x \to 5^+} \cos{\left(\pi x \right)} = -1
lim⁡x→∞cos⁡(πx)=⟨−1,1⟩\lim_{x \to \infty} \cos{\left(\pi x \right)} = \left\langle -1, 1\right\rangle
More at x→oo
lim⁡x→0−cos⁡(πx)=1\lim_{x \to 0^-} \cos{\left(\pi x \right)} = 1
More at x→0 from the left
lim⁡x→0+cos⁡(πx)=1\lim_{x \to 0^+} \cos{\left(\pi x \right)} = 1
More at x→0 from the right
lim⁡x→1−cos⁡(πx)=−1\lim_{x \to 1^-} \cos{\left(\pi x \right)} = -1
More at x→1 from the left
lim⁡x→1+cos⁡(πx)=−1\lim_{x \to 1^+} \cos{\left(\pi x \right)} = -1
More at x→1 from the right
lim⁡x→−∞cos⁡(πx)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\left(\pi x \right)} = \left\langle -1, 1\right\rangle
More at x→-oo
One‐sided limits [src] 
 lim cos(pi*x)
x->5+
lim⁡x→5+cos⁡(πx)\lim_{x \to 5^+} \cos{\left(\pi x \right)} 
-1
−1-1 
= -1.0
 lim cos(pi*x)
x->5-
lim⁡x→5−cos⁡(πx)\lim_{x \to 5^-} \cos{\left(\pi x \right)} 
-1
−1-1 
= -1.0
= -1.0
Numerical answer [src] 
-1.0
-1.0
The graph
Limit of the function cos(pi*x)
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