As x → π, cos x → ?
As x → (π/3)^+, cos x → ?
and this one please <3
As x → 0−, sin x →?
As x → (π/4), sin x → ?
well, you know that
cos π = -1
cos π/3 = 1/2
sin 0 = 0
sin π/4 = 1/√2
Since all these function are continuous everywhere, the limits from both sides exist, and are the same. Namely, the function value there.
To find the limit of a function as x approaches a certain value, we need to look at the behavior of the function as x gets arbitrarily close to that value.
As x approaches π (pi), the cosine function, cos x, approaches -1. Cosine is a periodic function that oscillates between -1 and 1 as x varies. At π, cos x equals -1. Therefore, as x approaches π from either side (from the left or the right), cos x will approach -1.
On the other hand, as x approaches π/3 from the positive side, cos x approaches 1/2. At π/3, cos x equals 1/2. As x gets arbitrarily close to π/3 while approaching from the positive side, cos x will approach 1/2.
In summary:
- As x approaches π, cos x approaches -1.
- As x approaches π/3 from the positive side, cos x approaches 1/2.