Question

Assertion :$\underset{x\to \infty }{lim}\frac{x^n+n{x}^{n-1}+1}{\left[x^n\right]}=0,n\in I$ (where $[.]$ represents greatest integer function). Reason: $x-1<\left[x\right]\le x$, (where $[.]$ represents greatest integer function).

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect and Reason is correct

Answer 1

The correct option is D Assertion is incorrect and Reason is correct

For all $x$, $x-1<\left[x\right]\le x$, where $[.]$ denotes greatest integer function.
$\Rightarrow x^n-1<\left[x^n\right]\le x^n\Rightarrow \frac{1}{x^n}\le \frac{1}{\left[x^n\right]}<\frac{1}{x^n-1}$
Multiplying the inequation by $x^n+n{x}^{n-1}+1$ and taking the limit as $x\to \infty$, we get,
$\underset{x\to \infty }{lim}\frac{x^n+n{x}^{n-1}+1}{x^n}\le \underset{x\to \infty }{lim}\frac{x^n+n{x}^{n-1}+1}{\left[x^n\right]}<\underset{x\to \infty }{lim}\frac{x^n+n{x}^{n-1}+1}{x^n-1}$
Evaluating the limits on the left and right side of the inequality, we obtain
$\underset{x\to \infty }{lim}\frac{x^n+n{x}^{n-1}+1}{x^n}=\underset{x\to \infty }{lim}\frac{x^n+n{x}^{n-1}+1}{x^n-1}=1$
And hence by sandwich theorem,
$\underset{x\to \infty }{lim}\frac{x^n+n{x}^{n-1}+1}{\left[x^n\right]}=1$
$\Rightarrow$ Assertion is incorrect.