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Section 2.7 : Limits at Infinity, Part I

1. For \(f\left( x \right) = 4{x^7} - 18{x^3} + 9\) evaluate each of the following limits.
   1. \(\mathop {\lim }\limits_{x \to \, - \infty } f\left( x \right)\)
   2. \(\mathop {\lim }\limits_{x \to \,\infty } f\left( x \right)\)
   Solution
2. For \(h\left( t \right) = \sqrt[3]{t} + 12t - 2{t^2}\) evaluate each of the following limits.
   1. \(\mathop {\lim }\limits_{t \to \, - \infty } h\left( t \right)\)
   2. \(\mathop {\lim }\limits_{t \to \,\infty } h\left( t \right)\)
   Solution

For problems 3 – 10 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. \(\displaystyle f\left( x \right) = \frac{{8 - 4{x^2}}}{{9{x^2} + 5x}}\) Solution
2. \(\displaystyle f\left( x \right) = \frac{{3{x^7} - 4{x^2} + 1}}{{5 - 10{x^2}}}\) Solution
3. \(\displaystyle f\left( x \right) = \frac{{20{x^4} - 7{x^3}}}{{2x + 9{x^2} + 5{x^4}}}\) Solution
4. \(\displaystyle f\left( x \right) = \frac{{{x^3} - 2x + 11}}{{3 - 6{x^5}}}\) Solution
5. \(\displaystyle f\left( x \right) = \frac{{{x^6} - {x^4} + {x^2} - 1}}{{7{x^6} + 4{x^3} + 10}}\) Solution
6. \(\displaystyle f\left( x \right) = \frac{{\sqrt {7 + 9{x^2}} }}{{1 - 2x}}\) Solution
7. \(\displaystyle f\left( x \right) = \frac{{x + 8}}{{\sqrt {2{x^2} + 3} }}\) Solution
8. \(\displaystyle f\left( x \right) = \frac{{8 + x - 4{x^2}}}{{\sqrt {6 + {x^2} + 7{x^4}} }}\) Solution