What is the limit as x approaches pi of cos(x+cos(x)) ?

The limit as x approaches pi of cos(x+cos(x)) is cos(pi-1)

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limit as x approaches pi of cos(x+cos(x))

x \to π lim (cos (x + cos (x)))

cos (π - 1)

Decimal

- 0.54030 \dots

Solution steps

x \to π lim (cos (x + cos (x)))

Plug in the value

x = π

= cos (π + cos (π))

Use the following trivial identity:

cos (π) = (- 1)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= cos (π - 1)

What is the limit as x approaches pi of cos(x+cos(x)) ?
The limit as x approaches pi of cos(x+cos(x)) is cos(pi-1)