You are given cos x=1−x22!+x44!−x66!......;
sin x=x−x33!+x55!−x77!......
tan x=x+x33+2.x515......
Find the value of limx→0x cosx+sinxx2+tanx
We can solve this problem using standard limits, which we will cover in the next topic. Here, we will show you how the standard expansions can be used to solve this problem.
We have,
sin x=x−x33!+x55!.....
cos x=1−x22!+x44!−x66!......
tan x=x+x33+2x515......
We will substitute these values in our expression and simplify to get:
⇒x cosx+sinxx2+tanx=x(1−x22!+x44!+..)+(x−x33!+x55!.....)x2+(x+x33+2x515......)
⇒limit=limx→0x[(1−x22!+x44!+..)+(1−x23!+x45!.....)]x(1+x+x23+2x415......)
=1+11
=21=2