Define Horizontal Asymptotes

We have learned how to read limits from a graph intuitively and identify horizontal asymptotes.




  • \(\displaystyle\lim_{x\to -\infty} f(x)=\)




  • \(\displaystyle\lim_{x\to \infty} f(x)=\)

How many horizontal asymptotes can we have in one graph? How about vertical asymptotes?

Using limits, how would you define a horizontal asymptote?

The line \(y=L\) is a horizontal asymptote of the curve \(y=f(x)\) if either \[\displaystyle \lim_{x\to -\infty} f(x) = L \quad \mbox{ or }\quad \lim_{x\to \infty} f(x) = L.\]

Evaluate Limits at Infinity

Example 1. \(\displaystyle \lim_{x\to\infty} \dfrac{\sin x}{x}.\)

Example 2. \(\displaystyle \lim_{x\to\infty} \sin\left(\frac{1}{x}\right).\)

Example 3. \(\displaystyle \lim_{x\to\infty} e^{1/x}.\)

Example \(\boldsymbol \pi\). \(\displaystyle \lim_{x\to \infty} \dfrac{2}{1 + e^{-x}}.\)

Example 4. \(\displaystyle \lim_{x\to - \infty} \dfrac{2}{1 + e^{-x}}.\)

Example 5. \(\displaystyle \lim_{x\to\infty} \dfrac{3 x^2 + x - 2}{5 x^2 + 4x + 1}.\)

Example 6. \(\displaystyle \lim_{x\to -\infty} \dfrac{2 - x + 5x^2}{ 3 x^2 + 1}.\)

Example 7. \(\displaystyle \lim_{x\to\infty} \dfrac{3 x^2 + x - 2}{x^3 + 1}.\)

Example 8. \(\displaystyle \lim_{x\to -\infty} \dfrac{x^3}{ (x-1)^2(x+1)^2}.\)

Example 9. \(\displaystyle \lim_{x\to \infty} \dfrac{\sqrt{2x^2 + 1}}{5x - 2}.\)

Example 10. \(\displaystyle \lim_{x\to -\infty} \dfrac{\sqrt{4x^2 - 1}}{2x + 3}.\)

Example 11. \(\displaystyle \lim_{x\to \infty} \left(\sqrt{x^2 + 1} - x\right).\)

Example 12. \(\displaystyle \lim_{x\to \infty} \left(\sqrt{4x^2 + x - 1} - 2x\right).\)

Example 13. \(\displaystyle \lim_{x\to \infty} \left(x^2 - 100x\right).\)

Example 14. \(\displaystyle \lim_{x\to -\infty} \left(\sqrt{x^2 + 1} + 3x\right).\)

Verify your answers graphically.