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

Question

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

Jaelyn Rosario · Accepted Answer

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 &quot;oscillating&quot; between +0 and -0 which you see is 0.

Mark Elliott · Accepted Answer

The limit is 0.To show this, use the squeeze (pinch, sandwich) theorem.  −  1  ≤  cos  ⁡  (      π    x    )  ≤  1 for   x  ≠  0Because       x          2        &amp;gt;  0 for all   x  ≠  0, we can multiply the inequality without reversing it:  −      x          2        ≤      x          2        cos  ⁡  (      π    x    )  ≤      x          2       for   x  ≠  0Note 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

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

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