Problem 21 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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 ∞.

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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 ∞.