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Chapter 2: Problem 20

Determine the following limits. $$\lim _{x \rightarrow-\infty}\left(3 x^{7}+x^{2}\right)$$

Short Answer

Expert verified
Answer: The limit of the function as x approaches negative infinity is \(-\infty\).

Step by step solution

01

Identify the dominant term

The given function is: $$f(x) = 3x^7 + x^2$$ In this polynomial function, the dominant term is the one with the highest exponent, which is \(3x^7\) in this case.
02

Analyze the behavior of dominant term

We can observe that the dominant term \(3x^7\) has an odd exponent, and its leading coefficient is positive. Thus, when x approaches negative infinity, the dominant term itself approaches negative infinity.
03

Compare the behavior of other terms to the dominant term

As x approaches negative infinity, the other term, \(x^2\), approaches positive infinity. However, its growth rate is significantly slower than that of the dominant term (\(3x^7\)) due to the smaller exponent. Therefore, we can essentially neglect the \(x^2\) term.
04

Determine the limit of the function

Since the dominant term dominates the behavior of the function as x approaches negative infinity, we can write: $$\lim_{x\rightarrow-\infty}(3x^7 + x^2) \approx \lim_{x\rightarrow-\infty}(3x^7)$$ Now, recognizing that the \(3x^7\) term approaches negative infinity when x approaches negative infinity, the limit of the function is: $$\lim_{x\rightarrow-\infty}(3x^7 + x^2) = -\infty$$

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Most popular questions from this chapter

Let $$g(x)=\left\\{\begin{array}{ll}x^{2}+x & \text { if } x<1 \\\a & \text { if } x=1 \\\3 x+5 & \text { if } x>1.\end{array}\right.$$ a. Determine the value of \(a\) for which \(g\) is continuous from the left at 1. b. Determine the value of \(a\) for which \(g\) is continuous from the right at 1. c. Is there a value of \(a\) for which \(g\) is continuous at \(1 ?\) Explain.

We say that \(\lim _{x \rightarrow \infty} f(x)=\infty\) if for any positive number \(M,\) there is \(a\) corresponding \(N>0\) such that $$f(x)>M \quad \text { whenever } \quad x>N$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{x^{2}+x}{x}=\infty$$

Analyzing infinite limits graphically Graph the function \(y=\tan x\) with the window \([-\pi, \pi] \times[-10,10] .\) Use the graph to analyze the following limits. a. \(\lim _{x \rightarrow \pi / 2^{+}} \tan x\) b. \(\lim _{x \rightarrow \pi / 2^{-}} \tan x\) d. \(\quad \lim _{n}\) tan \(x\) c. \(\lim _{x \rightarrow-\pi / 2^{+}} \tan x\) d. \(\lim _{x \rightarrow-\pi / 2^{-}} \tan x\)

Sketch a possible graph of a function \(f\) that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes. $$\lim _{x \rightarrow 0^{+}} f(x)=\infty, \lim _{x \rightarrow 0^{-}} f(x)=-\infty, \lim _{x \rightarrow \infty} f(x)=1$$, $$\lim _{x \rightarrow-\infty} f(x)=-2$$

Limits of composite functions. $$\begin{aligned}&\text { Suppose } g(x)=f(1-x), \text { for all } x, \lim _{x \rightarrow 1^{+}} f(x)=4, \text { and }\\\&\lim _{x \rightarrow 1^{-}} f(x)=6 . \text { Find } \lim _{x \rightarrow 0^{+}} g(x) \text { and } \lim _{x \rightarrow 0^{-}} g(x)\end{aligned}$$.
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