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Cosine limits Let n be a positive integer. Use graphical
Chapter 4, Problem 74RE(choose chapter or problem)
Let n be a positive integer. Use graphical and/ or analytical methods to verify the following limits:
a. \(\lim _{x \rightarrow 0}\ \frac{1-\cos x^n}{x^{2n}}=\frac{1}{2}\) b. \(\lim _{x \rightarrow 0}\ \frac{1-\cos^n x}{x^2}=\frac{n}{2}\)
Questions & Answers
QUESTION:
Let n be a positive integer. Use graphical and/ or analytical methods to verify the following limits:
a. \(\lim _{x \rightarrow 0}\ \frac{1-\cos x^n}{x^{2n}}=\frac{1}{2}\) b. \(\lim _{x \rightarrow 0}\ \frac{1-\cos^n x}{x^2}=\frac{n}{2}\)
ANSWER:Solution Step 1 In this problem we have to verify the given limits either by using graphical method or by using analytical method. Let us use analytical method to verify the given limits. 1cos x 1 a. lim x2n = 2 x0 To verify this limit we will be using the following trigonometric formula. 1. cos + sin = 1 2 2 2 2. cos 2 = cos sin n Now consider lim 1cos x x0 x2n By using 2., cos x can be written as n 2 xn 2 xn cos x = cos ( 2 sin (2) …. (i) And by using 1., 1 can be written as n n 1 = cos 2 ( ) + sin 2 ( ) …. (ii) 2 2 1cos xn Using (i) and ( ii), expand the numerator of lim x2n as follows x0 n cos2( ) +sin2 ( ) (cos2( ) sin2 ( )