Answer:
When we have a function like:
[tex]f(x) = \frac{g(x)}{h(x)}[/tex]
This function will have a discontinuity only if it diverges, and a divergence can happen when the denominator is equal to zero and the numerator is different than zero.
In this case, we have the equation:
[tex]f(x) = \frac{x^2 + 2*x}{x + 2}[/tex]
Here the denominator is:
h(x) = x + 2
This is equal to zero when:
x + 2 = 0
x = -2
Now we need to see what happens with the numerator when x = -2
g(-2) = (-2)^2 + 2*(-2) = 0
Is equal to zero.
Then we need to see the limit when x -> -2, and use the L'Hopital theorem.
[tex]\lim_{x \to \ - 2} \frac{x^2 + 2*x}{x + 2}[/tex]
Because we have zero over zero at that point, we need to look at the quotients of the derivatives of both numerator and denominator.
[tex]\lim_{x \to \ - 2} \frac{2*x+ 2}{ 2} = \frac{2*-2 + 2}{2} = -1[/tex]
Then the function does not diverge, then the function has no discontinuity.
We also could look at the graph of f(x) to see it:
Our function is a linear function, and this is because the numerator is x times the denominator, then the function is:
f(x) = x.
