How do you find the limit of # (1 - cos2Q) / (4Q^2)# as Q approaches 0?

Question

How do you find the limit of  # (1 - cos2Q) / (4Q^2)# as Q approaches 0?

Christopher Allers · Accepted Answer

#1/2#

#lim_{Q to 0} (1 - cos2Q) / (4Q^2)#

using the double angle identity

#= lim_{Q to 0} (1 - (color{red}{1 - 2 sin^2 Q})) / (4Q^2)#

#= lim_{Q to 0} (2 sin^2 Q) / (4Q^2)#

#= lim_{Q to 0} 1/2 ( sin Q) / (Q) (sin Q) / (Q)#

#= 1/2 * 1 * 1#

because #= lim_{z to 0} ( sin z) / (z) = 1# is a well known limit

Paisley Johnson · Answer

undefined To find the limit of (1 - cos^2Q) / (4Q^2) as Q approaches 0, we can use the limit definition.

First, we simplify the expression by using the trigonometric identity cos^2Q = 1 - sin^2Q.

(1 - cos^2Q) / (4Q^2) = (1 - (1 - sin^2Q)) / (4Q^2) = sin^2Q / (4Q^2)

Next, we can rewrite sin^2Q as (sinQ)^2.

(sinQ)^2 / (4Q^2)

Now, we can cancel out the common factor of Q^2 in the numerator and denominator.

(sinQ)^2 / (4Q^2) = (sinQ)^2 / 4

Finally, as Q approaches 0, sinQ also approaches 0. Therefore, the limit of (1 - cos^2Q) / (4Q^2) as Q approaches 0 is 0.