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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Limits

NDA 2021 · Mathematics · Limits

What is _ n arrow ∞ (a^n+b^n)/(a^n-b^n) where a > b > 1 , equal to?

What is lim⁡n→∞an+bnan−bn\lim _{n \rightarrow \infty} \frac{a^n+b^n}{a^n-b^n} where a>b>1a > b > 1, equal to?

Options

  1. A-1
  2. B0
  3. C1
  4. DLimit does not exist

Correct answer

C. 1

Step-by-step solution

lim⁡x→∞an+bnan−bnlim⁡x→∞an[1+(ba)n]an[1−(ba)n]a>b>1a>b⇒ab>1\begin{aligned} & \lim _{x \rightarrow \infty} \frac{a^n+b^n}{a^n-b^n} \\ & \lim _{x \rightarrow \infty} \frac{a^n\left[1+\left(\frac{b}{a}\right)^n\right]}{a^n\left[1-\left(\frac{b}{a}\right)^n\right]} \\ & a > b > 1 \\ & a > b \Rightarrow \frac{a}{b} > 1\end{aligned}a>b⇒ab<1=lim⁡x→∞[1+(ba)n][1−(ba)n]=1+01−0∴lim⁡x→∞an+bnan−bn=1\begin{aligned} & a > b \Rightarrow \frac{a}{b} < 1 \\ & =\lim _{x \rightarrow \infty} \frac{\left[1+\left(\frac{b}{a}\right)^n\right]}{\left[1-\left(\frac{b}{a}\right)^n\right]}=\frac{1+0}{1-0} \\ & \therefore \lim _{x \rightarrow \infty} \frac{a^n+b^n}{a^n-b^n}=1\end{aligned}$
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