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Chapter 2: Problem 21

Determine the following limits. $$\lim _{x \rightarrow-\infty}\left(-3 x^{16}+2\right)$$

Short Answer

Expert verified
Question: Determine the limit of the function f(x) = -3x^16 + 2 as x approaches negative infinity. Answer: The limit as x approaches negative infinity is 鈭�.

Step by step solution

01

Identify the dominant term

In the given function, the dominant term is the term with the highest exponent of x, which is -3x^16. As x approaches negative infinity, this term will be the most significant part of the function.
02

Decide whether to simplify or ignore the other terms

In this case, the other term in the function is 2, which does not have an x component. Since the function is dominated by -3x^16, we can ignore the 2.
03

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

The dominant term is -3x^16. Since the exponent is even, the term will always be positive regardless of the sign of x. As x becomes more negative, -3x^16 will become larger. Therefore, as x approaches negative infinity, -3x^16 will approach positive infinity.
04

Compute the limit of the function

Given that we ignored the other term (2) since its effect is insignificant compared to the dominant term (-3x^16) as x approaches negative infinity, we can determine the limit of the function: $$\lim _{x \rightarrow-\infty}\left(-3 x^{16}+2\right) = \lim _{x \rightarrow-\infty}(-3 x^{16}) = \infty$$ The limit of the function as x approaches negative infinity is 鈭�.

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