Limits at Infinity

Reading

• Sections 2.7

Practice problems

• Section 2.7: 1, 3, 7, 9, 13, 17, 19, 37
• In class: 2.7 35
• To turn in (due Wednesday, together with 2.8): 2.7 8, 20

Notes

Limits at Infinity

• We can consider limits when \(x\to\infty\) or \(x\to \infty\).
• If \(\displaystyle\lim_{x\to\infty}f(x) = L\), we say that the line \(y=L\) is a horizontal asymptote for \(f\).

• Key limits:

  For all \(n > 0\): \[\lim_{x\to\infty}x^n = \infty,\qquad \lim_{x\to \infty}x^{-n}=0\] For odd whole numbers \(n\): \[\lim_{x\to -\infty}x^n = -\infty\] For even whole numbers \(n\): \[\lim_{x\to -\infty}x^n = \infty\] For all whole numbers \(n\): \[\lim_{x\to -\infty}x^{-n} = 0\]

• Limits of rational functions behave in the same way as their leading terms:

  \[\lim_{x\to\pm\infty}\frac{a_n x^n + a_{n-1}x^{n-1}+\cdots + a_0}{b_m x^m + b_{m-1}x^{m-1}+\cdots + b_0} =\frac{a_n}{b_n}\lim_{x\to\pm\infty} x^{n-m}\]

• Example: \(\displaystyle\lim_{x\to\infty}\frac{x^3-2x+1}{2x^2+1}\)
• One approach to this: Factor out greatest power at numerator and denominator separately. Leaves a lot of \(\frac{1}{x^n}\) that go to \(0\).

• Another approach to limits at infinity: Set \(t=\frac{1}{x}\), look at \(t\to 0^+\) or \(t\to 0^-\).