How do you find the horizontal asymptote for #f(x) = (3x) / (x+4)#?

Answer 1

I found #y=3#

The horizontal asymptote is a line towards which the curve, described by your function, tends to get as near as possible.

To find it you can try to see what happens to your function when #x# becomes VERY big....and see if your functions "tends" to some kind of fixed value: as #x# becomes very big, say #x=1,000,000# you have: #f(1,000,000)=(3*1,000,000)/(1,000,000+4)# let us forget the #4# that is negligible compared to #1,000,000#; you have:

#f(1,000,000)=(3*cancel(1,000,000))/(cancel(1,000,000))=3#

So when #x# becomes very big positively (and negatively, you can try this) your functions "tends" to get near the value #3#! So the horizontal line of equation #y=3# will be your asymptote!

You can plot your function and see this tendency! graph{(3x)/(x+4) [-41.1, 41.07, -20.56, 20.53]}

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Answer 2

Serenity Johnson

To find the horizontal asymptote for the function $f(x) = \frac{3x}{x+4}$ , you need to examine the behavior of the function as $x$ approaches positive or negative infinity.

For rational functions like this one, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $y = 0$ .

In this case, the degree of the numerator is 1 and the degree of the denominator is also 1. To find the horizontal asymptote, divide the leading coefficient of the numerator by the leading coefficient of the denominator:

$\lim_{x \to \pm \infty} \frac{3x}{x+4} = \frac{3}{1} = 3$

So, the horizontal asymptote of $f(x)$ is $y = 3$ .

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.