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When the graph of a function f(x) is drawn, vertical asymptotes are points where the value of f(x) goes to `+-oo` .
Here is the graph of `f(x) = (x(x-7))/(x^3-49x)`
The vertical asymptotes lie at the values of x for which the denominator of the function is equal to 0.
f(x) = (x(x-7))/(x^3-49x)
= (x(x-7))/(x*(x^2 - 49))
= (x(x-7))/(x*(x - 7)(x+7))
= 1/(x + 7)
The denominator of the function is 0 at x = -7
The vertical asymptote of the function lies at x = -7.
Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function.
To find the vertical asymptotes of:` (x(x-7))/(x^3-49x)`
At first check the zeroes of the denominator.
The vertical asymptote is at x=-7 but not at x=0 and x=7. The reason is that at x=0 and x=7, the numerator is also zero, and thus, there will be a hole at x=0 and x=7.
Therefore, the vertical asymptote is at x=-7 .
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