A function is said to have a removable discontinuity at x=a if: 1. f is either n

Question

A function is said to have a removable discontinuity at x=a if:

1. f is either not defined or not continuous at x=a.

2. f(a) could either be defined or redefined so that the new function IS continuous at x=a.

Let f(x)={ ((7/x)+((-6x+14)/(x(x-2)))) if x (does not equal) 0 and x (does not equal) 2}

{ 6 , if x=0}

Show that f(x) has a removable discontinuity at x=0 and determine what value for f(0) would make f(x) continuous at x=0. Must redefine f(0)= _____?

Hint: Try combining the fractions and simplifying.
The discontinuity at x=2 is actually NOT a removable discontinuity, just in case you were wondering.

Explanation / Answer

Alright taking the hint: f(x) = 7(x-2)/x(x-2) + (-6x+14)/x(x-2) = (7x - 14 -6x +14) /x(x-2) = x/x(x-2) = 1/(x-2) Thus at x = 0 the function has a removable discontinuity because you could redefine the function such that f(0) = -1/2

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