limit of (cos x)/(x + 2) as x approaches 0

Question

find the limit analytically (direct substitution, factoring, conjugate, definition of derivative or lim of sin x/ x = 1 as x approaches 0

Answer 1

lim(x->0) cos(x) / (x + 2)
= lim(x->0) cos(x) / lim(x->0) (x + 2)
= cos(lim(x->0) x) / (lim(x->0) x + 2)
= cos(0) / (0 + 2)
= 1 / 2

Notice that sin(x) / x is of the 0 / 0 indeterminate form.
Thus, we can apply the L'Hopsital rule.

lim(x->0) sin(x) / x
= lim(x->0) cos(x) / 1 [By L'hospital rule]
= cos(lim(x->0) x) / 1
= cos(0) / 1
= 1 / 1
= 1

Comments

Wrong.
cos(0)=1
0+2=2
The function (cos(x))/(x+2) is continous at x=0 so you can plug in 0 into the equation and get the limit and the point value.
Thus,
cos(0)/(0+2)=1/2=.5