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Section 2.7 : Limits at Infinity, Part I

1. For \(f\left( x \right) = 8x + 9{x^3} - 11{x^5}\) evaluate each of the following limits.
   1. \(\mathop {\lim }\limits_{x \to \, - \infty } f\left( x \right)\)
   2. \(\mathop {\lim }\limits_{x \to \,\infty } f\left( x \right)\)
2. For \(h\left( t \right) = 10{t^2} + {t^4} + 6t - 2\) evaluate each of the following limits.
   1. \(\mathop {\lim }\limits_{x \to \, - \infty } f\left( x \right)\)
   2. \(\mathop {\lim }\limits_{x \to \,\infty } f\left( x \right)\)
3. For \(g\left( z \right) = 7 + 8z + \sqrt[3]{{{z^4}}}\) evaluate each of the following limits.
   1. \(\mathop {\lim }\limits_{x \to \, - \infty } f\left( x \right)\)
   2. \(\mathop {\lim }\limits_{x \to \,\infty } f\left( x \right)\)

For problems 4 – 17 answer each of the following questions.

(a) Evaluate \(\mathop {\lim }\limits_{x \to \, - \infty } f\left( x \right)\)

(b) Evaluate \(\mathop {\lim }\limits_{x \to \,\infty } f\left( x \right)\)

(c) Write down the equation(s) of any horizontal asymptotes for the function.

1. \(\displaystyle f\left( x \right) = \frac{{10{x^3} - 6x}}{{7{x^3} + 9}}\)
2. \(\displaystyle f\left( x \right) = \frac{{12 + x}}{{3{x^2} - 8x + 23}}\)
3. \(\displaystyle f\left( x \right) = \frac{{5{x^8} - 9}}{{{x^3} + 10{x^5} - 3{x^8}}}\)
4. \(\displaystyle f\left( x \right) = \frac{{2 - 6x - 9{x^2}}}{{15{x^2} + x - 4}}\)
5. \(\displaystyle f\left( x \right) = \frac{{5x + 7{x^4}}}{{4 - {x^2}}}\)
6. \(\displaystyle f\left( x \right) = \frac{{4{x^3} - 3{x^2} + 2x - 1}}{{10 - 5x + {x^3}}}\)
7. \(\displaystyle f\left( x \right) = \frac{{5 - {x^8}}}{{2{x^3} - 7x + 1}}\)
8. \(\displaystyle f\left( x \right) = \frac{{1 + 4\sqrt[3]{{{x^2}}}}}{{9 + 10x}}\)
9. \(\displaystyle f\left( x \right) = \frac{{25x + 7}}{{\sqrt {5{x^2} + 2} }}\)
10. \(\displaystyle f\left( x \right) = \frac{{\sqrt {8 + 11{x^2}} }}{{ - 9 - x}}\)
11. \(\displaystyle f\left( x \right) = \frac{{\sqrt {9{x^4} + 2{x^2} + 3} }}{{5x - 2{x^2}}}\)
12. \(\displaystyle f\left( x \right) = \frac{{6 + {x^3}}}{{\sqrt {8 + 4{x^6}} }}\)
13. \(\displaystyle f\left( x \right) = \frac{{\sqrt[3]{{2 - 8{x^3}}}}}{{4 + 7x}}\)
14. \(\displaystyle f\left( x \right) = \frac{{1 + x}}{{\sqrt[4]{{5 + 2{x^4}}}}}\)