Key takeaway: A limit at infinity is the value a function approaches as x increases or decreases without bound, and a finite result marks a horizontal asymptote.
Grade 12 – AP Calculus ABAP Calculus AB Unit 1Calculus
Definition of Limit at Infinity
A limit at infinity describes the long-term behavior of a function: what happens to the outputs as \(x\) becomes enormous, positive or negative. If the outputs settle near a number \(L\), the graph flattens along the horizontal line \(y=L\).
For a quotient of polynomials, compare the highest powers. When the bottom grows faster the outputs shrink to 0, when both grow equally fast the answer is the ratio of leading coefficients, and when the top grows faster the outputs run off to infinity. The reliable technique is to divide every term by the highest power of \(x\) found in the denominator.
Formula and notation
The two directions are written
\[\lim_{x\to\infty}f(x)=L,\qquad \lim_{x\to-\infty}f(x)=M.\]
The symbol \(\infty\) is not a number; it says that \(x\) grows without bound. For any positive power \(n\), \(\frac{1}{x^n}\) tends to 0.
Key facts
- A finite limit at infinity gives a horizontal asymptote at that height.
- Equal degrees give the ratio of leading coefficients.
- A bigger degree on the bottom forces the outputs toward 0.
- Left and right ends of the graph can have different horizontal asymptotes.
Limit at Infinity example
Find \(\lim_{x\to\infty}\dfrac{5x^2-3}{2x^2+8x}\) and \(\lim_{x\to\infty}\dfrac{4x+1}{x^2-3}\).
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First quotient: divide every term, top and bottom, by \(x^2\). That gives \(\frac{5-3/x^2}{2+8/x}\).
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As \(x\) grows, \(\frac{3}{x^2}\) and \(\frac{8}{x}\) both shrink to 0, leaving \(\frac{5}{2}\).
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Second quotient: the bottom has degree 2 and the top degree 1. Dividing by \(x^2\) gives \(\frac{4/x+1/x^2}{1-3/x^2}\), which heads to \(\frac{0}{1}=0\).
The first value is \(\frac{5}{2}\) and the second is 0.
Common mistake
Watch out
Sol sees both the top and the bottom of \(\frac{5x^2-3}{2x^2+8x}\) growing to infinity and says the quotient is \(\frac{\infty}{\infty}=1\).
Why it is wrong
The form \(\frac{\infty}{\infty}\) is indeterminate, not equal to 1. The top grows 5 units per \(x^2\) and the bottom grows 2 units per \(x^2\), so the ratio settles at \(\frac{5}{2}\). The table values 2.08 and 2.27 confirm the limit at infinity.
Related terms and lessons
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