Question

State the horizontal asymptote of the function f(x) = 3x over x + 5

Answer 1

so -5?

Answer 2

$$\frac{3x}{x+5}=\frac{3x+15-15}{x+5}=\frac{3(x+5)-15}{x+5}=\frac{3(x+5)}{x+5}-\frac{15}{x+5}=3-\frac{15}{x+5}$$

Answer 3

Observe that as we let \(x\to\infty\), i.e. it gets bigger and bigger, \(\dfrac{15}{x+5}\to0\) -- dividing by a bigger and bigger number will surely make the fraction smaller and smaller. The horizontal asymptote is precisely this sort-of behavior, observing what happens as we let \(x\to\pm\infty\).

Answer 4

As we can see, as \(x\) gets really really negative or really really positive, the \(\dfrac{15}{x+5}\) term doesn't do as much and we're really only focusing on that first term, \(3\). Hence we say we have a horizontal asymptote at \(y=3\) in the sense that our function behaves more and more like \(y=3\) as we let \(x=\pm3\) -- this is seen visually as the graph "hugs" the line \(y=3\) ever more tightly

Answer 5

er as we let \(x\to\pm\infty\) -- typo

Answer 6

sorry I gave you the vertical asymptote x = -5

Answer 7

Here's a plot where you can sorta see what I'm talking about: http://www.wolframalpha.com/input/?i=y%3D3x%2F%28x%2B5%29%2C+y%3D3+for+-50000%3C%3Dx%3C%3D50000

Notice that in the case we have \(\dfrac{ax+b}{cx+d}\), for example, we observe asymptotic behavior like \(\frac{a}c\) -- we take the ratio of the coefficients of the highest degree \(x\) terms. This also works for any rational function \(\dfrac{P(x)}{Q(x)}\) where \(P,Q\) are polynomial functions of the same degree

Answer 8

In the case \(P\) is of higher degree than \(Q\), e.g. \(\dfrac{x^3}{x^2+2}\), we have no horizontal asymptote -- the numerator ends up growing a lot faster in the long run so the function doesn't tend to approach any horizontal asymptote, instead it goes off to \(\pm\infty\). If \(P\) is of *lesser* degree, however, we observe the opposite behavior -- the denominator overtakes our numerator in the long run and the whole thing gets smaller, towards \(0\).

Answer 9

Thank you c:

Answer 10

For example, \(\dfrac{x+700}{x^2}\) -- our denominator is of higher degree than our numerator so it outpaces the numerator in the long run and our fraction tends towards \(0\) as \(x\to\pm\infty\) -- check the graph!
http://www.wolframalpha.com/input/?i=%28x%2B700%29%2Fx%5E2