Question

The value of $\underset{x\to n}{lim}\frac{a^x+b^x}{a^x-b^x}$, ( where $a>b>0$) is

• A. $1$
• B. $-1$
• C. $\frac{1}{2}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

The correct option is A $1$

Given

$\underset{x\to n}{lim}\frac{a^x+b^x}{a^x-b^x}$

$\underset{x\to n}{lim}\frac{a^x(1+{\left(\frac{b}{a}\right)}^x)}{a^x(1-{\left(\frac{b}{a}\right)}^x)}$

$\underset{x\to n}{lim}\frac{1+{\left(\frac{b}{a}\right)}^x}{1-{\left(\frac{b}{a}\right)}^x}$

$a>b⟹\frac{b}{a}<1⟹{\left(\frac{b}{a}\right)}^n\sim 0$

$⟹\frac{1+{\left(\frac{b}{a}\right)}^n}{1-{\left(\frac{b}{a}\right)}^n}=1$