# Calculus_微积分_Limits at Infinity

## Calculus_微积分_Examples of limits at infinity

The evaluate-approximate-render (EAR) procedure is used for this type of limit as well.
Example 1 Determine $\lim x \rightarrow \infty \frac{2 x+8}{x^3+4 x+2}$. Solution We evaluate the expression at $x=\Omega$ :
$$\lim x \rightarrow \infty \frac{2 x+8}{x^3+4 x+2}=\frac{2 \Omega+8}{\Omega^3+4 \Omega+2} \quad \approx \frac{2 \Omega}{\Omega^3}=\frac{2}{\Omega^2}=2 \omega^2$$
The limit is 0 .
The graph of the function with the limit that is taken in example 1 has a horizontal asymptote of $y=0$ (on the right).
Example 2 Evaluate $\lim {x \rightarrow-\infty} \frac{x^2+4}{3 x^2+5 x-9}$ Solution We evaluate the expression at $x=-\Omega$ : $$\lim x \rightarrow-\infty \frac{x^2+4}{3 x^2+5 x-9}=\frac{(-\Omega)^2+4}{3(-\Omega)^2+5(-\Omega)-9}=\frac{\Omega^2+4}{3 \Omega^2-5 \Omega}$$ The limit is $\frac{1}{3}$. The graph of the function with the limit that is taken in example 2 has a horizontal asymptote of $y=\frac{1}{3}$ (on the left). Example 3 Find the limit: $\lim x \rightarrow \infty \frac{5 x^3-2 x+7}{x^2+7 x-1}$ Solution We evaluate the expression at $x=\Omega$ : $$\lim {x \rightarrow \infty} \frac{5 x^3-2 x+7}{x^2+7 x-1}=\frac{5 \Omega^3-2 \Omega+7}{\Omega^2+7 \Omega-1} \approx \frac{5 \Omega^3}{\Omega^2}=5 \Omega \doteq \infty$$
The limit is $\infty$.

## Calculus_微积分_Examples of finding asymptotes

Determining whether a function has a horizontal asymptote (or two) is accomplished by checking the limits at infinity. Since $\lim x \rightarrow \infty f(x)$ may be different from $\lim x \rightarrow-\infty f(x)$, both limits should be checked.
Example 5 Find all horizontal asymptotes on the graph of $f(x)=$ $\frac{3 x-7}{4 x+\sqrt[3]{x^3+5 x}}$
Solution We first check the right side by evaluating the limit as $x \rightarrow \infty$
$$\lim {x \rightarrow \infty} \frac{3 x-7}{4 x+\sqrt[3]{x^3+5 x}}=\frac{3 \Omega-7}{4 \Omega+\sqrt[3]{\Omega^3+5 \Omega}} \quad \approx \frac{3 \Omega}{4 \Omega+\sqrt[3]{\Omega^3}}=\frac{3}{4 \Omega}$$ The function has a horizontal asymptote on the right, $y=\frac{3}{5}$. We also need to check the left side to see if the result is different. We therefore evaluate the limit as $x \rightarrow-\infty$ : $$\lim {x \rightarrow-\infty} \frac{3 x-7}{4 x+\sqrt[3]{x^3+5 x}}=\frac{3(-\Omega)-7}{4(-\Omega)+\sqrt[3]{(-\Omega)^3+5(-\Omega)}} \quad=\frac{}{-4 \Omega}$$
The function has a horizontal asymptote of $y=\frac{3}{5}$ on the left side as well.

Reading Exercise 15 Find any horizontal asymptotes on the graph of $y=\frac{4 x-1}{x+3}$

# 微积分代考

## Calculus_微积分_Examples of limits at infinity

$$\lim x \rightarrow \infty \frac{2 x+8}{x^3+4 x+2}=\frac{2 \Omega+8}{\Omega^3+4 \Omega+2} \approx \frac{2 \Omega}{\Omega^3}=\frac{2}{\Omega^2}=2 \omega^2$$

$$\lim x \rightarrow-\infty \frac{x^2+4}{3 x^2+5 x-9}=\frac{(-\Omega)^2+4}{3(-\Omega)^2+5(-\Omega)-9}=\frac{\Omega^2+4}{3 \Omega^2-5 \Omega-}$$

$$\lim x \rightarrow \infty \frac{5 x^3-2 x+7}{x^2+7 x-1}=\frac{5 \Omega^3-2 \Omega+7}{\Omega^2+7 \Omega-1} \approx \frac{5 \Omega^3}{\Omega^2}=5 \Omega \doteq \infty$$

## Calculus_微积分_Examples of finding asymptotes

$$\lim x \rightarrow \infty \frac{3 x-7}{4 x+\sqrt[3]{x^3+5 x}}=\frac{3 \Omega-7}{4 \Omega+\sqrt[3]{\Omega^3+5 \Omega}} \approx \frac{3 \Omega}{4 \Omega+\sqrt[3]{\Omega^3}}$$

$$\lim x \rightarrow-\infty \frac{3 x-7}{4 x+\sqrt[3]{x^3+5 x}}=\frac{3(-\Omega)-7}{4(-\Omega)+\sqrt[3]{(-\Omega)^3+5(-\Omega)}}=$$

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