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$\lim_{x\to\infty} \left( \frac{x+a}{2} - \sqrt{\left(\frac{x+a}{2}\right)^2 - abx} \right)$

Rearrange the function

$\left( \frac{x+a}{2} - \sqrt{\left(\frac{x+a}{2}\right)^2 - abx} \right) \left( \frac{\frac{x+a}{2} + \sqrt{\left(\frac{x+a}{2}\right)^2 - abx}}{\frac{x+a}{2} + \sqrt{\left(\frac{x+a}{2}\right)^2 - abx}} \right)$ $\frac { \left(\frac{x+a}{2}\right)^2 - \left(\frac{x+a}{2}\right)^2 + abx } { \frac{x+a}{2} + \sqrt{\left(\frac{x+a}{2}\right)^2 - abx} }$ $\frac { abx } { \frac{x+a}{2} + \sqrt{\left(\frac{x+a}{2}\right)^2 - abx} }$

Back to taking the limit

$\lim_{x\to\infty} \left( \frac { abx } { \frac{x+a}{2} + \sqrt{\left(\frac{x+a}{2}\right)^2 - abx} } \right)$

LHopitals Rule

$\lim_{x\to\infty} \left( \frac { \frac{d}{dx} abx } { \frac{d}{dx} \frac{x+a}{2} + \frac{d}{dx} \sqrt{\left(\frac{x+a}{2}\right)^2 - abx} } \right)$ $\lim_{x\to\infty} \left( \frac { ab } { \frac{1}{2} + \frac{\frac{x+a}{2} - ab}{2\sqrt{\left(\frac{x+a}{2}\right)^2 - abx}} } \right)$

Moving the limit inside

$\frac { ab } { \frac{1}{2} + \lim \left( \frac{\frac{x+a}{2} - ab}{2\sqrt{\left(\frac{x+a}{2}\right)^2 - abx}} \right) }$

Lets look at the limit in the denominator

$\lim \left( \frac{\frac{x+a}{2} - ab}{2\sqrt{\left(\frac{x+a}{2}\right)^2 - abx}} \right)$

Again apply LHopitals Rule

$\lim \left( \frac{ \frac{1}{2} } {2 \frac{d}{dx} \sqrt{\left(\frac{x+a}{2}\right)^2 - abx}} \right)$

Move the limit inside

$\frac{ \frac{1}{2} } { 2 \lim \left( \frac{d}{dx} \sqrt{\left(\frac{x+a}{2}\right)^2 - abx} \right) }$ $\frac{ \frac{1}{2} } { 2 \lim \left( \frac{\frac{x+a}{2} - ab}{2\sqrt{\left(\frac{x+a}{2}\right)^2 - abx}} \right) }$

Notice that we now have a repeating fraction

$\lim \left( \frac{\frac{x+a}{2} - ab}{2\sqrt{\left(\frac{x+a}{2}\right)^2 - abx}} \right) = \frac{ \frac{1}{2} } { 2 \lim \left( \frac{\frac{x+a}{2} - ab}{2\sqrt{\left(\frac{x+a}{2}\right)^2 - abx}} \right) }$ $y = \frac{ \frac{1}{2} } {2 y }$ $y = \frac{1}{2}$

Apply this to the original limit

$\frac { ab } { \frac{1}{2} + \frac{1}{2} } = ab$

Charles Rymal

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