Question

Let f, g and h are functions having a common domain D and $h\left(x\right)\le f\left(x\right)\le g\left(x\right),x\in D$ and if $

\displaystyle \lim_{x\rightarrow a}f(x)=l$. This is known as Sandwich Theorem. Using the result, compute the following limits.

The value of $\underset{x\to 0}{lim}\frac{\left|x\right|}{\sqrt{x^4+4x^2+7}}$

• A. $1$
• B. $0$
• C. $\frac{1}{2}$
• D. doesn't exist

Answer 1

The correct option is B $0$

Since $\sqrt{x^4+4x^2+7}\ge 1$, so
$0\le \frac{|x|}{\sqrt{x^4+4x^2+7}}\le |x|$. But ${lim}_{x\to 0}|x|=0$
Hence $0\le \underset{x\to 0}{lim}\frac{|x|}{\sqrt{x^4+4x^2+7}}\le \underset{x\to 0}{lim}|x|=0$
$\therefore \underset{x\to 0}{lim}\frac{|x|}{\sqrt{x^4+4x^2+7}}=0$