The hole appears in the rational function when the numerator and the denominator have the same zeroes
Since the rational function is
[tex]h(x)=\frac{x+7}{x^2-49}[/tex]
Factorize the denominator
[tex]x^2-49=(x+7)(x-7)[/tex]
The rational function h(x) is
[tex]h(x)=\frac{x+7}{(x+7)(x-7)}[/tex]
Since (x + 7) is in both numerator and denominator
Then there is a hole at x + 7 = 0
Let us find the value of x
[tex]\begin{gathered} x+7=0 \\ x+7-7=0-7 \\ x=-7 \end{gathered}[/tex]
The whole is at x = -7
Then simplify the fraction to find the value of y at x = -7
[tex]h(x)=\frac{(x+7)}{(x+7)(x-7)}[/tex]
Cancel the bracket (x+7) up by the same bracket down
[tex]h(x)=\frac{1}{x-7}[/tex]
Substitute x by -7
[tex]\begin{gathered} h(-7)=\frac{1}{-7-7} \\ h(-7)=\frac{1}{-14} \\ y=-\frac{1}{14} \end{gathered}[/tex]
The hole is at (-7, -1/14)
