Problem 24 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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

Step 1: Find the limit of the first term

The first term of the function is \(2x^{-8}\). To find the limit as x approaches negative infinity, we need to determine how this term behaves when x decreases: $$\lim_{x \rightarrow -\infty} 2x^{-8} = \lim_{x \rightarrow -\infty} \frac{2}{x^8}$$ As x approaches negative infinity, the denominator \(x^8\) will approach infinity, so the fraction becomes smaller and smaller. Consequently: $$\lim_{x \rightarrow -\infty} \frac{2}{x^8} = 0$$

Step 2: Find the limit of the second term

The second term of the function is \(4x^3\). To find the limit as x approaches negative infinity, we need to determine how this term behaves when x decreases: $$\lim_{x \rightarrow -\infty} 4x^3$$ As x approaches negative infinity, \(x^3\) will also approach negative infinity, so the entire term will approach negative infinity: $$\lim_{x \rightarrow -\infty} 4x^3 = -\infty$$

Step 3: Find the overall limit of the function

Now that we have the limits of both terms, we can find the overall limit of the function: $$\lim _{x \rightarrow-\infty}\left(2 x^{-8}+4 x^{3}\right) = \lim_{x \rightarrow -\infty} 2x^{-8} + \lim_{x \rightarrow -\infty} 4x^3$$ Since we found that the limit of the first term is 0 and the limit of the second term is \(-\infty\), the overall limit of the function is: $$\lim _{x \rightarrow-\infty}\left(2 x^{-8}+4 x^{3}\right) = 0 + (-\infty) = -\infty$$