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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 does not exist (DNE).

Step by step solution

01

Identify the leading term

The leading term of the function is the term with the highest power of x, which in this case is \(3x^7\). This term will dominate the behavior of the function as x approaches negative infinity.
02

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

As x approaches negative infinity, the leading term will also approach negative infinity because x is raised to an odd power, and a negative number raised to an odd power remains negative. We can express this as $$\lim_{x \rightarrow -\infty}(3x^7) \rightarrow -\infty$$
03

Analyze the behavior of the rest of the terms

Now we need to analyze the behavior of the other terms in the function. In our case, there is only one additional term: \(x^2\). As x approaches negative infinity, \(x^2\) approaches positive infinity because a negative number squared becomes positive. $$\lim_{x \rightarrow -\infty}(x^2) \rightarrow \infty$$
04

Combine the limits of the separate terms

We can now find the limit of the function as x approaches negative infinity by combining the limits of the separate terms: $$\lim_{x \rightarrow -\infty}(3x^7 + x^2) = \lim_{x \rightarrow -\infty}(3x^7) + \lim_{x \rightarrow -\infty}(x^2)$$ Since \(\lim_{x \rightarrow -\infty}(3x^7)=-\infty\) and \(\lim_{x \rightarrow -\infty}(x^2)=\infty\), this expression becomes: $$-\infty + \infty$$
05

Determine the limit

The expression \(-\infty + \infty\) is not well-defined because adding positive infinity and negative infinity does not provide a specific value. Therefore, we cannot determine the limit as x approaches negative infinity for this function. The limit does not exist, which we can denote as DNE (does not exist). $$\lim_{x \rightarrow -\infty}\left(3 x^{7}+x^{2}\right) = DNE$$

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