Question

Which of the following must be true for the graph of the function
f(x)= (x^2-25)/5x-25)
There is:
I. a removable discontinuity at x=5
II. a vertical asymptote at x=5
III. an infinite discontinuity at x=5

possibles answers:
a) I only
b) II only
c) III only
d) I, II, III

Thanks in advance

Answer 1

The denominator cannot be equal to zero, and it is possible when the x-value is,
5x - 25 = 0
5x = 25
x = 5

This is however a removable discontinuity; there is a common factor in the numerator and denominator and therefore the function can be simplified as,
f(x) = (x^2-25)/(5x-25)
f(x) = (x+5)(x-5) / (5)(x-5)
f(x) = (x+5) / 5
where x = all real numbers except 5.