Problem 19 - Calculus Early Transcendentals

Question

Determine the following limits. $$\lim _{x \rightarrow \infty}\left(3 x^{12}-9 x^{7}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Identify the leading term

The given polynomial function is: \(3x^{12} - 9x^{7}\). The leading term is the term with the highest power of x, which in this case is \(3x^{12}\). Step 2 - Analyze the behavior of the leading term as x approaches infinity

Step 2: Analyze the behavior of the leading term

As x approaches infinity, the term \(3x^{12}\) will grow significantly faster than \(-9x^{7}\). This means that, in the limit, the value of the function will be determined by the behavior of \(3x^{12}\) alone. Step 3 - Find the limit of the leading term as x approaches infinity

Step 3: Find the limit of the leading term

The limit of the leading term as x approaches infinity can be found separately. So, $$\lim_{x \rightarrow \infty} 3x^{12} = \infty$$ Step 4 - Determine the limit of the given function as x approaches infinity

Step 4: Determine the limit of the given function

Since the leading term \(3x^{12}\) dominates the function as x approaches infinity, the limit of the given function is the limit of the leading term. Therefore, $$\lim_{x \rightarrow \infty} (3x^{12} - 9x^{7}) = \lim_{x \rightarrow \infty} 3x^{12} = \infty$$