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cos(x)+sin(x)
Limit of the function/ cos(x)+sin(x)

Limit of the function cos(x)+sin(x)

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 lim  (cos(x) + sin(x))
   pi                  
x->--+                 
   6
lim⁡x→π6+(sin⁡(x)+cos⁡(x))\lim_{x \to \frac{\pi}{6}^+}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) 
Limit(cos(x) + sin(x), x, pi/6)
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
-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.02-2
Plot the graph
Other limits x→0, -oo, +oo, 1
lim⁡x→π6−(sin⁡(x)+cos⁡(x))=12+32\lim_{x \to \frac{\pi}{6}^-}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = \frac{1}{2} + \frac{\sqrt{3}}{2}
More at x→pi/6 from the left
lim⁡x→π6+(sin⁡(x)+cos⁡(x))=12+32\lim_{x \to \frac{\pi}{6}^+}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = \frac{1}{2} + \frac{\sqrt{3}}{2}
lim⁡x→∞(sin⁡(x)+cos⁡(x))=⟨−2,2⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = \left\langle -2, 2\right\rangle
More at x→oo
lim⁡x→0−(sin⁡(x)+cos⁡(x))=1\lim_{x \to 0^-}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = 1
More at x→0 from the left
lim⁡x→0+(sin⁡(x)+cos⁡(x))=1\lim_{x \to 0^+}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = 1
More at x→0 from the right
lim⁡x→1−(sin⁡(x)+cos⁡(x))=cos⁡(1)+sin⁡(1)\lim_{x \to 1^-}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = \cos{\left(1 \right)} + \sin{\left(1 \right)}
More at x→1 from the left
lim⁡x→1+(sin⁡(x)+cos⁡(x))=cos⁡(1)+sin⁡(1)\lim_{x \to 1^+}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = \cos{\left(1 \right)} + \sin{\left(1 \right)}
More at x→1 from the right
lim⁡x→−∞(sin⁡(x)+cos⁡(x))=⟨−2,2⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = \left\langle -2, 2\right\rangle
More at x→-oo
Rapid solution [src] 
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1   \/ 3 
- + -----
2     2
12+32\frac{1}{2} + \frac{\sqrt{3}}{2}
Expand and simplify
One‐sided limits [src] 
 lim  (cos(x) + sin(x))
   pi                  
x->--+                 
   6
lim⁡x→π6+(sin⁡(x)+cos⁡(x))\lim_{x \to \frac{\pi}{6}^+}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) 
      ___
1   \/ 3 
- + -----
2     2
12+32\frac{1}{2} + \frac{\sqrt{3}}{2} 
= 1.36602540378444
 lim  (cos(x) + sin(x))
   pi                  
x->---                 
   6
lim⁡x→π6−(sin⁡(x)+cos⁡(x))\lim_{x \to \frac{\pi}{6}^-}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) 
      ___
1   \/ 3 
- + -----
2     2
12+32\frac{1}{2} + \frac{\sqrt{3}}{2} 
= 1.36602540378444
= 1.36602540378444
Numerical answer [src] 
1.36602540378444
1.36602540378444
The graph
Limit of the function cos(x)+sin(x)
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