Limit when $y>>a$ of a derived solution

Question

I am able to do part d), however I am very stuck on part e).

If $y >> a$ then surely we get

$\phi(x,y)$ $= \frac{1}{\pi} \Big[ tan^{-1} \Big( \frac{x+a}{y}\Big)-tan^{-1} \Big(\frac{x-a}{y}\Big)\Big] \rightarrow \frac{1}{\pi} \Big[ tan^{-1} \Big( \frac{x}{y}\Big)-tan^{-1} \Big(\frac{x}{y}\Big)\Big]=0$

which clearly is not correct

Answer 1

Hint

The classical formulas are $$\tan ^{-1}(a)+\tan ^{-1}(b)=\tan ^{-1} \big( \frac {a+b} {1-a~b}\big)$$ $$\tan ^{-1}(a)-\tan ^{-1}(b)=\tan ^{-1} \big( \frac {a-b} {1+a~b}\big)$$

I am sure that you can take from here.