Find the limit of a trigonometric function?

Question

lim x--> 0 sin x / x + sin x

I'm not sure how to find the limit of the problem written above. We're only in the second week of class so we haven't learned the L'Hospital rule or how to derive.

Answer 1

Thiis is 0/0 indeterminate form. Because we can epply L'Opital Rule: lim x → 0 of sinx/(x + sinx) =

lim x → 0 of d/dx(sinx)/d/dx(x + sinx) = lim x → 0 of cosx/(1 + cosx) = cos0/(1 + cos0) = 1/(1 + 1) = 1/2

Answer 2

Brittany

Do it numerically.

Set your calculator to radians.

Then calculate this ratio for values of x closer and closer to 0.

For example

x sin x / (x + sin x)

1 0.4569

0.5 0.4895

0.1 0.4996

0.01 0.4999958

0.001 0.49999995

So as x approaches zero from the right, looks like the limit of the function will be 0.5

And you need to do the same from the left, so try x = -1, -0.5, -0.1, -0.01, -0.001 and see what happens to the function value. You should find the same (as x gets closer to 0 function gets closer to 0.5)

So right hand limit = left hand limit = 0.5

So the limit you need is 0.5. In calculus the limit does not need to be actually attained, you just have to be able to get as close to it as you like.

Mike

Answer 3

limit of sinx/x = 1

take the derivative of numerator and denominator to get cosx/1,

derivative of sinx is cosx, derivative of x is 1. then let x=0 to get cos(0)/1 = 1/1 = 1

limit of sinx at 0 is just sin(0) = 0

limit of sinx/x + sinx = 1+0 = 1