Section 2.6: Limits at Infinity; Horizontal Asymptotes

1) Find the limit \( \displaystyle{\lim_{x \to \infty} \frac{x^2-5x+7}{4x^2-x^3+1}} \)   solution

2) Find the limit \( \displaystyle{\lim_{x \to \infty} \frac{3x^4-2x+5}{4x^3-5x+1}}\)  solution

3) Find the limit \( \displaystyle{\lim_{x \to -\infty} \frac{1-2x+4x^3}{2+x^2-3x^3}}\)  solution

4) Find the limit \(\displaystyle{\lim_{x \to \infty} \frac{3x^2-2x+5}{4x^2 + 2x+1}}\)  solution

5) Find the limit \(\displaystyle{\lim_{x \to \infty} \frac{\sqrt{9x^4+5}}{5x^2 -3x+1}}\)  solution

6) Evaluate \( \displaystyle{\lim_{x \to \infty} \frac{1+e^x}{1-3e^x}} \)   solution

7) Find the limit \( \displaystyle{\lim_{x \to -\infty} \frac{2x}{\sqrt{x^2-1}}}\)  solution

8) Find the horizontal and vertical asymptotes of the curve.    \(\displaystyle{y=\frac{5+x^{3}}{x-x^{3}}}\)  solution

9) Find the horizontal and vertical asymptotes of the curve.    \(\displaystyle{f(x)=\frac{\sqrt{2 x^{2}+1}}{5x-3}}\)  solution

 

10. Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions.   solution

\( \quad \displaystyle{\lim _{x \rightarrow 3} f(x)=-\infty, \: \lim _{x \rightarrow \infty} f(x)=2, \: \lim _{x \rightarrow-\infty} f(x)=0, \: \lim _{x \rightarrow 0^{+}} f(x)= \infty, \:\lim _{x \rightarrow 0^{-}} f(x)=-\infty} \)

 

11. Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions.   solution

\( \quad \displaystyle{ f(0)=3,   \lim_{x \to 0^-}   f(x)=4,   \lim_{x \to 0^+}   f(x)= 2,}\) \(\displaystyle{ \lim_{x \to -\infty}   f(x)= -\infty,   \lim_{x \to 4^-}   f(x)=-\infty,}\) \(\displaystyle{ \lim_{x \to 4^+}   f(x)= \infty,   \lim_{x \to \infty}   f(x)=3 } \)

 

12. Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions.  solution

 \(\quad \displaystyle{ f(2)=0, \, \lim_{x \to 2^-} f(x)=\infty, \, \lim_{x \to 2^+} f(x)=-\infty, \, \lim_{x \to \infty} f(x)= -1, \, \lim_{x \to -\infty} f(x)= 0} \)

 

13. Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions.  solution

\( \quad \displaystyle{f(0) =0,  \, \lim_{x \to -1 ^-} f(x)=\infty, \,  \lim_{x \to -1 ^+} f(x)=-\infty, \, \lim_{x \to 1 ^-} f(x)=-\infty, \, \lim_{x \to 1 ^+} f(x)=\infty, \, \lim_{x \to \infty} f(x)= 1, \, \lim_{x \to -\infty} f(x)= -2} \)

 

14. Evaluate the following limits.    solution

\( \qquad \displaystyle{ \lim_{x\to \infty} \arctan(x^2) \qquad \qquad \qquad \lim_{x\to -\infty} \arctan(e^x) }\)