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Use the Squeeze Theorem to show that  lim x→0 x2 cos(16?x) = 0.

Use the Squeeze Theorem to show that  lim x→0 x2 cos(16?x) = 0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A graphing calculator is recommended.
Use the Squeeze Theorem to show that 
lim x→0 x2 cos(16?x) = 0.
Illustrate by graphing the functions 
f(x) = −x2,
 
g(x) = x2 cos(16?x),
 and 
h(x) = x2
 on the same screen.
Let 
f(x) = −x2, g(x) = x2 cos(16?x),
 and 
h(x) = x2.
 Then 
     ≤ cos(16?x) ≤     
   ⇒   
     ≤ x2 cos(16?x) ≤      .
 Since 
lim x→0 f(x) = lim x→0 h(x) =  ,
 by the Squeeze Theorem we have 
lim x→0 g(x) =  .
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