12 December 2021

Statement

Find the value of the following limit.

\(\lim\limits_{x \to \infty} \frac{x^{7} - 1}{x^{6} + 1}\)

Solution

\(\lim\limits_{x \to \infty} \frac{x^{7} - 1}{x^{6} + 1}\)

The higher degree polynomial in the numerator indicates this limit will diverge since \(x^7 \gt x^6\) when \(x \to \infty\).

\(= \lim\limits_{x \to \infty} \frac{x^{7}(1 - \frac{1}{x^{7}})}{x^{6}(1 + \frac{1}{x^{6}})} \)

\(= \lim\limits_{x \to \infty} \frac{x(1 - \frac{1}{x^{7}})}{1 + \frac{1}{x^{6}}} \)

Since the denominator \(1 + \frac{1}{x^{6}} \neq 0\) when \(x \to \infty\) we can use the following limit property .

\(= \frac{\lim\limits_{x \to \infty} x(1 - \frac{1}{x^{7}})}{\lim\limits_{x \to \infty} 1 + \frac{1}{x^{6}}} \)

\(= \frac{\lim\limits_{x \to \infty} x \times \lim\limits_{x \to \infty}(1 - \frac{1}{x^{7}})}{\lim\limits_{x \to \infty} 1 + \frac{1}{x^{6}}} \)

\(= \frac{\infty \times 1}{1 + 0}\)

\(= \infty\)