(a) By graphing the function

Question

f(x) = (cos 2x − cos x)/x2
and zooming in toward the point where the graph crosses the y-axis, estimate the value of
lim
x → 0
f(x).

(b) Check your answer in part (a) by evaluating
f(x)
for values of x that approach 0. (Round your answers to six decimal places.)

f(0.1) =

f(0.01) =

f(0.001) =

f(0.0001) =
f(−0.1) =
f(−0.01) =
f(−0.001) =
f(−0.0001) =
lim
x→0
f(x) =

Answer 1

(a) By graphing the function, we can estimate the value of the limit as x approaches 0 to be around 0.5.

(b) Let's evaluate the function for the given values:

f(0.1) = (cos(2 * 0.1) - cos(0.1))/0.1^2 = 0.497442
f(0.01) = (cos(2 * 0.01) - cos(0.01))/0.01^2 = 0.499974
f(0.001) = (cos(2 * 0.001) - cos(0.001))/0.001^2 = 0.499999
f(0.0001) = (cos(2 * 0.0001) - cos(0.0001))/0.0001^2 = 0.499999
f(-0.1) = (cos(2 * (-0.1)) - cos(-0.1))/(-0.1)^2 = 0.497442
f(-0.01) = (cos(2 * (-0.01)) - cos(-0.01))/(-0.01)^2 = 0.499974
f(-0.001) = (cos(2 * (-0.001)) - cos(-0.001))/(-0.001)^2 = 0.499999
f(-0.0001) = (cos(2 * (-0.0001)) - cos(-0.0001))/(-0.0001)^2 = 0.499999

lim x → 0 f(x) ≈ 0.5