# Where have I gone wrong in evaluating $\lim_{x \to \infty}\sqrt x(\sqrt{x+c}- \sqrt x )$?

Evaluate $$\lim_{x \to \infty}\sqrt x(\sqrt{x+c}- \sqrt x )$$

Attempt:

\begin{align} \lim_{x \to \infty}\sqrt x(\sqrt{x+c}- \sqrt x ) &= \lim_{x\to \infty }(\sqrt{x^2+cx}- x) \\ &= \lim _{x\to \infty}x\left(\sqrt{\left(1+\dfrac{c}{x}\right)}-1\right)\\ &= \lim _{x \to \infty} x \times 0 \\ &= 0 \times \infty \\ &=0 \end{align}

But the answer given is :

$$\frac c 2$$

• you can mouseover the pinkish field to reveal – Vinyl_cape_jawa Apr 27 '18 at 9:05
• The issue with your reasoning is that you cannot keep one $x$ while making the other term $0$. The two terms go together. The same flawed reasoning would show $1 = \lim_{x \to \infty} x \frac{1}{x} = \lim_{x \to \infty} x 0 = 0$. It makes no sense to throw the limit onto only one term. – mathworker21 Apr 27 '18 at 9:12

Write like this (using $a^2-b^2=(a-b)(a+b)$)

$$\sqrt{x^2+cx} - x = \frac{ x^2+cx - x^2}{\sqrt{x^2+cx}+x} = \frac{cx}{\sqrt{x^2+cx}+x}$$

Now, factor $x$ numerator and denominator cancel and we obtain

$$\frac{ c }{ \sqrt{ 1 + \frac{c}{x} } + 1 }$$

• What have I done wrong? That was the question. – Archer Apr 27 '18 at 9:07
• $0*\infty$ is not a valid thing – Frank Moses Apr 27 '18 at 9:08
• you have to make simplification to make it a valid thing if it is possible – Frank Moses Apr 27 '18 at 9:08
• and I think $0\times \infty$ is not equal to $0$ – Frank Moses Apr 27 '18 at 9:09
• why not @FrankMoses – Archer Apr 27 '18 at 9:17

Taylor expansion of $\sqrt{1+f(x)}$ where $f(x)\to 0$ is:

$$\sqrt{1+f(x)}=1+\frac12f(x)+ o(f(x))$$ Then

$$\lim _{x\to \infty}x\left(\sqrt{1+\dfrac{c}{x}}-1\right)=\lim _{x\to \infty}x\left(1+\frac12\dfrac{c}{x}-1\right)=\frac c2$$

Note $$\displaystyle \lim_{x \to \infty} x = \infty$$ and $$\displaystyle \lim_{x \to \infty} \frac cx = 0$$ while

$$\displaystyle \lim_{x \to \infty} x \cdot\frac cx = c$$

1) Let $c=0,$ and $x \gt 0$.

The limit is?

2) Let $x > |c|>0.$

$f(x):= x^{1/2}[(x+c)^{1/2}-x^{1/2}] =$

$x^{1/2} \dfrac {c}{(x+c)^{1/2} +x^{1/2}}=$

$\dfrac{c}{(1+c/x)^{1/2} +1}.$

The limit is?