Calculus question?

Question

The value of lim (sin x – sin π) / (x - π) is
x→π


Please show me how to get to the answer, and not just the answer. Thank you.

Answer 1

Since the form of the limit is the indeterminate 0/0, you can use L'Hopital's Rule:

lim[x->π] (sin x - sin π) / (x - π) =

lim[x->π] [(d/dx)(sin x - sin π)] / [(d/dx)(x - π)] =

lim[x->π] cos x = -1

Answer 2

=> lim (sin x – sin π) / (x - π)
x-> π
=> lim (sin x – sin π) / (x - π)
x- π -> 0
Substitue x-π = y
=> lim [sin (y+π )– sin π]/ y
y-> 0
Let me tell u 2 theorems
sin(A+B)=sin A cos B + cos A sin B
lim sin y/ y = 1
y-> 0

By applying above 2 theorems
=> lim [sin (y+π )– sin π]/ y
y-> 0
=> lim [sin y cos π + cos y sin π - sin π]/y
y-> 0
But sin π = 0 and cos π = -1
Hence the equation becomes
=> lim [sin y (-1) + 0 - 0]/y
y-> 0
= -1 * lim sin y/ y
y-> 0

= -1

Hope i have put on all steps

Answer 3

As sin pi = 0 it is the limit of
sinx /(x-pi) x=>pi.
if you put x-pi =z
you get lim sin(pi+z) /z with z=>0= lim (sin pi cosz+cos pi sinz)/z = -sinz/z as cos pi =-1
It is known that lim sin z /z with z=>0 is 1
so the limit is -1

Answer 4

By definition this is the derivative of the sine function at pi, and so this is cos(pi) = -1.

But I don't know if you're allowed to assume this...