How do you find the limit of sinx/(x+sinx) as x->0?

1 Answer
Dec 8, 2016

Use L'Hôpital's rule and evaluate the resulting expression at 0.

Explanation:

Given: lim_(xto0)sin(x)/(x + sin(x)) = ?

Because the expression evaluated at 0, is the indeterminate form, 0/0, the use of L'Hôpital's rule is warranted.

Compute the derivative of the numerator:

(d(sin(x)))/dx = cos(x)

Compute the derivative of the denominator:

(d(x + sin(x)))/dx = 1 + cos(x)

Take the limit of the new fraction:

lim_(xto0)cos(x)/(1 + cos(x)) = cos(0)/(1 + cos(0)) = 1/2

According to L'Hôpital's rule, the limit of the original function goes to the same value:

lim_(xto0)sin(x)/(x + sin(x)) = 1/2