Question

Use a table of values to estimate the value of the limit.

\(\mathop {\lim }\limits_{x \to 1} \frac{{{x^6} - 1}}{{{x^{10}} - 1}}\)

Short answer

The value of the given function \(\mathop {lim}\limits_{x \to 1} \frac{{{x^6} - 1}}{{{x^{10}} - 1}}\) is \(\frac{3}{5}\).

Step-by-step solution

Introduction

The function does not exist on the given limit but it can sustain on the nearby values of the limit. To find the value of the limit we have to find the nearby limit values.

Given information

The given function is \(\mathop {lim}\limits_{x \to 1} \frac{{{x^6} - 1}}{{{x^{10}} - 1}}\).

Table of values

By putting the different values of x the\(\mathop {lim}\limits_{x \to 1} \frac{{{x^6} - 1}}{{{x^{10}} - 1}}\)will give the following table,

x	f(x)	x	f(x)
0.5	0.985337	1.5	0.183369
0.9	0.719397	1.1	0.484119
0.95	0.660186	1.05	0.540783
0.99	0.612018	1.01	0.588022
0.999	0.601200	1.001	0.598800

It seems the value of the limit

\(\begin{array}{c}\mathop {lim}\limits_{x \to 1} \frac{{{x^6} - 1}}{{{x^{10}} - 1}} = 0.6\\ = \frac{3}{5}\end{array}\)

Hence, the required value is \(\frac{3}{5}\).