Horizontal Asymptotes

Oblique Asymptotes

Vertical Asymptotes

Exercises - Domain and Asymptotes

1. Consider the function $f(x) = \frac{1}{x^2-1}$ a) Find domain and asymptotes (if there are)

2. Consider the function $f(x) = \frac{x^2-3}{x-2}$ a) Find domain and asymptotes (if there are)

3. Consider the function $f(x) = xe^{1/x}$. a) Determine the domain and asymptotes (if there are).

4. Find domain and asymptotes of the following functions and collect all the information on the cartesian plan: a) $f(x) = \frac{e^x-1}{e^x+1}$ b) $f(x) = \frac{-2x^4+2x^3+x^2-1}{x^3}$ c) $f(x) = \sqrt{4+x^2}$ d) $f(t) = t + \sqrt{4t^2-1}$ e) $f(x) = x-2+\frac{x^2}{\sqrt{x^2+9}}$

5. Find the domain and asymptotes of the following functions: a) $f(x) = x + \frac{1}{x}$ b) $f(x) = \frac{1-x-x^2}{x-1}$ c) $f(x) = \sqrt{x^2+1}$ d) $f(x) = \frac{1}{\ln x}$ e) $f(x) = e^t+1$ f) $f(x) = \arctan\left(\frac{x+2}{x-1}\right)$ g) $f(x) = \frac{1}{x\ln(x+1)}$

6. Consider the function $f(x) = \frac{x-1}{x^2+1}$. Mark true or false in the following sentences: a) The function has a vertical asymptote. b) The function has an oblique asymptote.

7. Consider the function $f(x) = \frac{4-5x^2}{x^2+x-2}$. Mark true or false in the following sentences: a) The function has a vertical asymptote. b) The function has an oblique asymptote.

8. Consider the function $f(x) = \frac{\sqrt{x}-x^2}{x}$. Mark true or false in the following sentences: a) $f$ has a vertical asymptote. b) $f$ has an oblique asymptote.

9. Consider the function $f(x) = \frac{x^4-10x^2+9}{x^3}$. Mark true or false in the following sentences: a) $f$ has a vertical asymptote. b) $f$ has an oblique asymptote.

10. Consider the function $f(x) = xe^{\frac{x-1}{x}}$. a) The function has a vertical asymptote. b) The function has an oblique asymptote.

11. For $f(x) = \frac{x-2}{x^2+2}$, analyze vertical asymptotes.

12. Apply the asymptotic comparison test for $\lim\limits_{x\to\infty} \frac{f(x)}{\frac{1}{x}} = \lim\limits_{x\to\infty} \frac{x(x-2)}{x^2+2} = 1$.

13. Consider the function $f(x) = e^{\frac{x}{x-1}}$ a) Find domain and asymptotes (if there are)

14. Consider the function $f(x) = \frac{1}{x^2-1}$ a) Find domain and asymptotes (if there are)

15. Consider the function $f(x) = xe^{1/x}$ a) Determine the domain and asymptotes (if there are)

16. Consider the function $f(x) = x^3 - 3x^2 + x - 3$ a) Determine the domain and asymptotes (if there are)

17. Find the domain and asymptotes of the following functions: a) $\frac{1}{x}-\sqrt{x}$ b) $\frac{s2^s}{2s+1}$ c) $\sqrt{x}-\sqrt{x-1}+x$ d) $\frac{1}{x\ln(x+1)}$ e) $x + \frac{1}{x}$