asymptotes of f(x)= x/(x(x+7))
1) Let's find the asymptotes of xx·(x + 7), for real x
Vertical asymptotes: find the excluded values, where the denominator is zero. Then check whether the curve goes to infinity near these values
Let's solve in ℝ: x·(x + 7) = 0
A product is zero if and only if one of its factors is zero: we solve each factor equal to 0
⇥1.1) Let's solve in ℝ: x = 0
⇥Solution found: x = 0
⇥1.2) Let's solve in ℝ: x + 7 = 0
⇥Solution found: x = -7
As x approaches -7 from the left (smaller values), the function tends to -∞; from the right (larger values), it tends to +∞
The function tends to infinity on at least one side: the line x = -7 is therefore a vertical asymptote
As x approaches 0 from the left (smaller values), the function tends to 17; from the right (larger values), it tends to 17
The limits at 0 are finite: this is a hole, not a vertical asymptote
At infinity, the numerator degree is smaller than the denominator degree: the line to test is y = 0
For a horizontal asymptote, the function must tend to a finite value: check the limit of xx·(x + 7) at both infinities
As x tends to +∞, xx·(x + 7) tends to 0; as x tends to -∞, xx·(x + 7) tends to 0
The horizontal asymptote is y = 0; there is no slant asymptote
▶Asymptotes:   { x = -7, y = 0 }
🖼/draw?f1=x%2F%28x*%28x+%2B+7%29%29&v=x&x2=-7&f3=0&b=-17.0&e=3.0&min=-0.59&max=0.49000000000000005&dash=2,3