Limits

This can be solved without using L-H rule

Let

$L\quad =\quad \underset { x\rightarrow 1 }{ lim } \quad \frac { P }{ 1-x^{ P } } \quad -\quad \frac { Q }{ 1-x^{ Q } } \quad \quad \longrightarrow (1)$ .

Now Replace

$x\quad \leftrightarrow \quad \frac { 1 }{ x }$ .

So That limit Doesn't Change.

$L\quad =\quad \underset { x\rightarrow 1 }{ lim } \quad \frac { Px^{ P } }{ x^{ P }-1 } \quad -\quad \frac { Qx^{ Q } }{ x^{ Q }-1 } \\ L\quad =\quad \underset { x\rightarrow 1 }{ lim } \quad -\quad \frac { Px^{ P } }{ 1-x^{ P } } \quad +\quad \frac { Qx^{ Q } }{ 1-x^{ Q } } \quad \longrightarrow (2)$ .

Now Add Two limit's

$\therefore \quad 2L=\quad P-Q\\ \Longrightarrow \quad L=\frac { P-Q }{ 2 }$ .

Q.E.D

by taking L.C.M and putting the limit we get $\frac{0}{0}$ form

so L.H can be applied

therefore we get

= $\huge{\frac{PQ( x^{P-1} - x^{Q -1})}{Px^{P-1}( x^{Q} -1) + Qx^{Q-1}( x^{P} -1)}}$

= $\huge{\frac{PQ(( x^{P-1}-1) -( x^{Q -1}-1))}{Px^{P-1}( x^{Q} -1) + Qx^{Q-1}( x^{P}-1)}}$

= $\huge{\frac{PQ(\frac{( x^{P-1}-1)}{x-1} -\frac{( x^{Q -1}-1)}{x-1})}{Px^{P-1}\frac{( x^{Q} -1)}{x-1} + Qx^{Q-1}\frac{( x^{P} -1)}{x-1}}}$

we know $\frac{x^{n} -1}{x-1}$ = $x + x^{2} + x^{3} + ............ + x^{n }$ ( sum of G.P)

therefore by keeping the limit we get

= $\frac{PQ( P -Q)}{2PQ}$

= $\frac{ P -Q}{2}$