#### Evaluate given limit: $\lim _{x \rightarrow \infty}\left(\sqrt{\left(23 x^2+23 x-3\right)}-\sqrt{\left(23 x^2-3\right)}\right)$Option: 1 $\frac{\sqrt{23}}{4}$Option: 2 $\frac{\sqrt{23}}{2}$Option: 3 $\frac{\sqrt{3}}{2}$Option: 4 $\frac{\sqrt{3}}{4}$

For solving the limits of the type ∞ - ∞, we simply rationalize the given limit, that is multiplying and dividing the limit by its additive inverse.

Additive inverse of $\lim _{x \rightarrow \infty}\left(\sqrt{\left(23 x^2+23 x-3\right)}-\sqrt{\left(23 x^2-3\right)}\right)$ would be $\lim _{x \rightarrow \infty}\left(\sqrt{\left(23 x^2+23 x-3\right)}+\sqrt{\left(23 x^2-3\right)}\right)$

Hence,

\begin{aligned} & =\lim _{x \rightarrow \infty} \frac{\left(\sqrt{\left(23 x^2+23 x-3\right)}-\sqrt{\left(23 x^2-3\right)}\right)\left(\sqrt{\left(23 x^2+23 x-3\right)}+\sqrt{\left(23 x^2-3\right)}\right)}{\left(\sqrt{\left(23 x^2+23 x-3\right)}+\sqrt{\left(23 x^2-3\right)}\right)} \\ = & \lim _{x \rightarrow \infty} \frac{\left(\left(23 x^2+23 x-3\right)-\left(23 x^2-3\right)\right)}{\left(\sqrt{\left(23 x^2+23 x-3\right)}+\sqrt{\left(23 x^2-3\right)}\right)} \\ = & \lim _{x \rightarrow \infty} \frac{23 x}{\left(\sqrt{\left(23 x^2+23 x-3\right)}+\sqrt{\left(23 x^2-3\right)}\right)} \end{aligned}$\text { Dividing the numerator and denominator by } \mathrm{x} \text { : }$

\begin{aligned} & =\lim _{x \rightarrow \infty} \frac{23}{\left(\sqrt{\left(23+\frac{23}{x}-\frac{3}{x^2}\right)}+\sqrt{\left.\left(23-\frac{3}{x^2}\right)\right)}\right.} \\ & =\frac{23}{2 \sqrt{23}} \\ & =\frac{\sqrt{23}}{2} \end{aligned}