Limit at infinity
The value a function approaches as x grows without bound. For fractions of polynomials, divide top and bottom by the highest power of x.
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Analogy
Watching a car drive away: it looks smaller and smaller and settles towards the horizon.
Example
Divide top and bottom by x^2.(3 + 1/x^2) / (1 + 5/x^2).As x grows, 1/x^2 tends to 0.Limit is 3 / 1 = 3.
In practice
Used to find horizontal asymptotes and to compare how fast different functions grow.
Practice
Flashcards
- What is the limit of (3x^2 + 1)/(x^2 + 5) as x tends to infinity?3, the ratio of the leading coefficients.
Questions people ask
How do I find the limit of a fraction as x goes to infinity?
Divide every term by the highest power of x in the denominator. Terms like 1/x and 1/x squared then vanish. If top and bottom have the same degree, you get the ratio of leading coefficients. If the top has higher degree, the limit is infinite.
What is a horizontal asymptote?
It is a horizontal line y = L that the graph approaches as x goes to plus or minus infinity. It comes from a limit at infinity. For (3x^2 + 1)/(x^2 + 5), the limit is 3, so the line y = 3 is a horizontal asymptote.