Expert: Paul Klarreich - 5/3/2007

Question
let f be defined as follows, where a not equal to 0, f(x)={x^2-a^2)/(x-a) for x not equal a, and 0, for x=a. which of the following are true about f?
1). lim as x approaches a f(x) exists
2). f(a) exists
3). f(x) is continuous at x=a

Answer
Questioner:   bhavika
Category:  Calculus
Private:  No
 
Subject:  calculus
Question:  let f be defined as follows, where a not equal to 0, f(x)={x^2-a^2)/(x-a) for x not equal a, and 0, for x=a. which of the following are true about f?
1). lim as x approaches a f(x) exists
2). f(a) exists
3). f(x) is continuous at x=a
...................................
Hi, Bhavika,

This is a direct application of the definition of continuity to a function, which I'll write like this:
************** USE COURIER FONT TO VIEW THIS **********
      | (x^2-a^2)/(x-a),  when  x /= a
f(x) = |
      | 0,                when  x = a

Now the questions:

1) lim as x approaches a f(x) exists
  
For this, you assume  x is NOT equal to a.  Then you take the top line:
    x^2 - a^2
lim  --------- =
x->a   x - a

     (x - a)(x + a)
lim   -------------- =
x->a      x - a

lim  x + a  = a + a = 2a
x->a

The limit EXISTS, and is equal to 2a.
..............

2). f(a) exists

Of course it does.  It says so right in the definition.  f(a) = 0.
...............
3). f(x) is continuous at x=a

No, it is not.  The definition of continuity requires that the limit and the function value must be the same.  They were not.