What are the removable and non-removable discontinuities, if any, of #f(x)=(2x^2+4x-70)/(x-5)#?

1 Answer
Jim H
Nov 2, 2015

#f# has a removable discontinuity at #5#.

Explanation:

#f(x)=(2x^2+4x-70)/(x-5)# is a rational function, so it is continuous on its domain which is all real numbers except #5#.

#f# has a discontinuity at #5#.

#lim_(xrarr5)f(x)= lim_(xrarr5)(2x^2+4x-70)/(x-5)#

# = lim_(xrarr5)(2(x-5)(x+7))/(x-5)#

# = lim_(xrarr5)2(x+7) = 2(5+7)#

So the limit at #5# exists, therefore the discontinuity at #5# is removable.

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