how do you evaluate a function with the squeeze theorem?

Question

 I need to evaluate the limit of  [(x^2-16)(x-4/|x-4|)  as x approaches 4 with the squeeze theorem. How do I go about doing this?

Answer 1

You can do it without the Squeeze Theorem:

 

When x < 4, x-4 < 0.  So, l x-4 l = -(x-4)

 

  lim_x→4^- [(x²-16)((x-4) / l x-4 l] = lim_x→4^- [(x²-16)(-1)] = 0

 

When x > 4,  x-4 > 0.  So, l x-4 l = x-4.

 

  lim_x→4⁺ [(x²-16)((x-4) / l x-4 l] = lim_x→4⁺ [(x²-16)(1)] = 0

 

Since the one-sided limits are both equal to zero, the value of the given limit is also zero.