How do you prove that the function #x*(x-2)/(x-2)# is not continuous at x=2?

Question

Ezra Adams · Accepted Answer

When #x=2# the expression #x*(x-2)/(x-2)# would require division by #0# (which is not defined).

For a function #f(x)# to be continuous at a point #x=a#, three conditions must be met: Since for #f(x)=x*(x-2)/(x-2)# does not meet condition 1 when #x=2#, it is not continuous at this point. Note that it is possible to define a function similar to this which is continuous: #f(x){(=x*(x-2)/(x-2),,("if "x!=2)),(=x,,("if "x=2)):}#

Cameron Alexander · Answer

undefined To prove that the function f(x) = x*(x-2)/(x-2) is not continuous at x=2, we can show that the limit of f(x) as x approaches 2 does not exist or is not equal to f(2).

First, let's simplify the function by canceling out the common factor (x-2) in the numerator and denominator:

f(x) = x*(x-2)/(x-2) = x

Now, let's evaluate the limit of f(x) as x approaches 2:

lim(x→2) x = 2

Since the limit of f(x) as x approaches 2 is equal to 2, we need to check if f(2) is also equal to 2.

f(2) = 2

Since the limit and the value of the function at x=2 are equal, we can conclude that the function f(x) = x*(x-2)/(x-2) is continuous at x=2.