lim x -> pi/2 (sin(x/2)-cos(x/3))
Evaluate
Please help
y=sin (PI/4)-cos(pi/6)
y= -.159
check that, I don't see it as an indeterminant form.
To evaluate the limit of the function `f(x) = sin(x/2) - cos(x/3)` as `x` approaches `pi/2`, we can use the limit properties and trigonometric identities.
Let's break it down step by step:
Step 1: Substitute the limit value into the function.
- Replace `x` with `pi/2` in the function:
`f(pi/2) = sin(pi/4) - cos(pi/6)`
Step 2: Apply the trigonometric identities:
- We know that `sin(pi/4) = cos(pi/4) = sqrt(2)/2` and `cos(pi/6) = sqrt(3)/2`.
- Therefore, substituting these values into the function:
`f(pi/2) = sqrt(2)/2 - sqrt(3)/2`
Step 3: Simplify the expression:
- To subtract the two fractions, we need to have a common denominator.
- Multiply the first fraction by `sqrt(3)/sqrt(3)` and the second fraction by `sqrt(2)/sqrt(2)` to get the common denominator of `2sqrt(6)/2`:
`f(pi/2) = (sqrt(2)/2)*(sqrt(3)/sqrt(3)) - (sqrt(3)/2)*(sqrt(2)/sqrt(2))`
`f(pi/2) = (sqrt(6)/2) - (sqrt(6)/2)`
`f(pi/2) = 0`
Therefore, the evaluation of the limit `lim x → pi/2 (sin(x/2) - cos(x/3))` is equal to `0`.