Question

determine whether the limit is equal to ∞ or -∞.
lim(x→∞) x² - 6

choose the correct answer below.

lim(x→∞) x² - 6 = ∞
lim(x→∞) x² - 6 = -∞

Explanation:

Step1: Analyze the function as x approaches infinity

As \(x\to\infty\), the term \(x^{2}\) dominates the function \(y = x^{2}-6\). Since \(x^{2}\) is a quadratic function with a positive leading - coefficient (\(a = 1>0\)), when \(x\) gets larger and larger, \(x^{2}\) also gets larger and larger.
The value of \(x^{2}\) goes to infinity as \(x\to\infty\), and subtracting 6 from a very large positive number still results in a very large positive number.

Step2: Determine the limit

We have \(\lim_{x\to\infty}(x^{2}-6)=\lim_{x\to\infty}x^{2}-\lim_{x\to\infty}6\). Since \(\lim_{x\to\infty}x^{2}=\infty\) and \(\lim_{x\to\infty}6 = 6\), then \(\lim_{x\to\infty}(x^{2}-6)=\infty\).

Answer:

\(\lim_{x\to\infty}x^{2}-6=\infty\)