Question

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

\(\mathop {\lim }\limits_{x \to 0} \frac{{\tan 3x}}{{\tan 5x}}\)

Short answer

The required value for the given function \(\mathop {\lim }\limits_{x \to 0} \frac{{\tan 3x}}{{\tan 5x}}\) is \(0.59\).

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 limit we have to find the nearby limit values.

Given information

The given function is, \(\mathop {\lim }\limits_{x \to 0} \frac{{\tan 3x}}{{\tan 5x}}\).

Table of values

By putting the different values of x the\(\mathop {\lim }\limits_{x \to 0} \frac{{\tan 3x}}{{\tan 5x}}\)will give the following table,

x	f(x)	x	f(x)
\( - 0.2\)	0.439279	0.2	0.439279
\( - 0.1\)	0.566236	0.1	0.566236
\( - 0.05\)	0.591893	0.05	0.591893
\( - 0.01\)	0.599680	0.01	0.599680
\( - 0.001\)	0.599997	0.001	0.599997

Both the limits are equal.

\(\mathop {\lim }\limits_{x \to 0} \frac{{\tan 3x}}{{\tan 5x}} = 0.59\)

Hence the limit is \(0.59\).