Question

The \[\mathop {\lim }\limits_{n \to \infty } \dfrac{{{a^n} + {b^n}}}{{{a^n} - {b^n}}}\], where \[a > b > 1\], is equal to?
A. -1
B. 1
C. 0
D. None of these

Answer 1

Hint: In order to solve this question first, we have to use the ratio of both terms given in inequality and find the higher power of that term. Then we take \[{a^n}\] common from numerator and denominator. Then cancel that factor from the equation and then put the limit and use the identified relation and simplify that to get the final answer.

Complete step by step answer:
We have given a relation \[a > b > 1\], from here we will separate \[a,b\] from this relation.
\[a > b\] as \[a,b\] is greater than 1. So, dividing by \[a\] both sides.
\[\dfrac{a}{a} > \dfrac{b}{a}\]
After further calculations.
\[\dfrac{b}{a} < 1\]
\[\Rightarrow {\left( {\dfrac{b}{a}} \right)^\infty } = 0\] because \[\dfrac{b}{a} < 1\]

Let the value of the limit is \[x\]. Now solving the given limit
\[x = \mathop {\lim }\limits_{n \to \infty } \dfrac{{{a^n} + {b^n}}}{{{a^n} - {b^n}}}\]
Taking \[{a^n}\] part common from numerator and denominator.
\[x = \mathop {\lim }\limits_{n \to \infty } \dfrac{{{a^n}\left( {1 + \dfrac{{{b^n}}}{{{a^n}}}} \right)}}{{{a^n}\left( {1 - \dfrac{{{b^n}}}{{{a^n}}}} \right)}}\]
Now canceling the common terms from numerator and denominator.
\[x = \mathop {\lim }\limits_{n \to \infty } \dfrac{{\left( {1 + \dfrac{{{b^n}}}{{{a^n}}}} \right)}}{{\left( {1 - \dfrac{{{b^n}}}{{{a^n}}}} \right)}}\]

Now putting the value of n in the expression and eliminating the expression of limit.
\[x = \dfrac{{\left( {1 + \dfrac{{{b^\infty }}}{{{a^\infty }}}} \right)}}{{\left( {1 - \dfrac{{{b^\infty }}}{{{a^\infty }}}} \right)}}\]
Now putting the relations \[{\left( {\dfrac{b}{a}} \right)^\infty } = 0\] in the final equation.
\[x = \dfrac{{\left( {1 + 0} \right)}}{{\left( {1 - 0} \right)}}\]
On simplifying this expression
\[\therefore x = 1\]
Hence, the value of the given limit \[\mathop {\lim }\limits_{n \to \infty } \dfrac{{{a^n} + {b^n}}}{{{a^n} - {b^n}}}\] is 1.

Therefore, option ‘B is the correct answer.

Note: To solve this question, students must know that if we divide an inequality by a positive number and that number is greater than one then inequality remains the same. And if we divide that inequality by a positive number and that is smaller than 1 then inequality changes its sign. We are able to use all arithmetic operators in the limit function and there are many rules related to limits like addition, subtraction, multiplication, and division.