How do you find the limit of x^2cos(pi/x) as x approaches 0?

Max Macias

Max Macias

Open question

2022-08-14

How do you find the limit of x 2 cos ( π x ) as x approaches 0?

Answer & Explanation

Jaelyn Rosario

Jaelyn Rosario

Beginner2022-08-15Added 16 answers

cos ( 1 / x ) oscillates between 1 and -1 really fast when aproaching to 0 but it always is a finite number. If you are multipliying this oscillating function to x 2 which modullates the oscillation it will oscillate between x 2 and x 2 .
So if you make the limit going to 0 it will "oscillating" between +0 and -0 which you see is 0.
Mark Elliott

Mark Elliott

Beginner2022-08-16Added 6 answers

The limit is 0.
To show this, use the squeeze (pinch, sandwich) theorem.
1 cos ( π x ) 1 for x 0
Because x 2 > 0 for all x 0, we can multiply the inequality without reversing it:
x 2 x 2 cos ( π x ) x 2 for x 0
Note that
lim x 0 ( x 2 ) = 0 = lim x 0 x 2 ,
So the squeeze theorem (or whatever you call it) tells us that
lim x 0 x 2 cos ( π x ) = 0

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