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Chapter 2: Problem 24

Determine the following limits. $$\lim _{x \rightarrow-\infty}\left(2 x^{-8}+4 x^{3}\right)$$

Short Answer

Expert verified
As x approaches negative infinity, the limit of the function $$\lim _{x \rightarrow-\infty}\left(2 x^{-8}+4 x^{3}\right)$$ is negative infinity or $$-\infty$$.

Step by step solution

01

Identify the dominant term

In our function $$f(x)=2 x^{-8}+4 x^{3}$$, we have two terms: one with power -8 and the other with power 3. As x approaches negative infinity, the term with the higher power will have a more significant impact on the overall value of the function. In this case, the term with higher power is $$4 x^{3}$$.
02

Analyze the behavior of the dominant term as x approaches negative infinity

As x approaches negative infinity, the value of the dominant term, $$4 x^{3}$$, will also approach negative infinity since the power of x is an odd number, and a negative number raised to an odd power results in a negative value.
03

Determine the limit

Since the dominant term is $$4 x^{3}$$, and its value approaches negative infinity when x approaches negative infinity, we can conclude that $$\lim _{x \rightarrow-\infty}\left(2 x^{-8}+4 x^{3}\right)$$ is also negative infinity. So, the final answer is: $$\lim _{x \rightarrow-\infty}\left(2 x^{-8}+4 x^{3}\right) = -\infty$$

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Most popular questions from this chapter

Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\). Then state the horizontal asymptote(s) of \(f\). Confirm your findings by plotting \(f\). $$f(x)=\frac{3 e^{x}+e^{-x}}{e^{x}+e^{-x}}$$

a. Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\), evaluate \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{x^{2}-9}{x(x-3)}$$

The limit at infinity \(\lim _{x \rightarrow \infty} f(x)=L\) means that for any \(\varepsilon>0,\) there exists \(N>0\) such that $$|f(x)-L|<\varepsilon \quad \text { whenever } \quad x>N$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{2 x+1}{x}=2$$

a. Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\), evaluate \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{x-1}{x^{2 / 3}-1}$$

a. Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\), evaluate \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{\sqrt{16 x^{4}+64 x^{2}}+x^{2}}{2 x^{2}-4}$$
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