What is the limit as x approaches p(x) of (g(x){x}-g(x){p(x)})/(x-p(x)) ?

The limit as x approaches p(x) of (g(x){x}-g(x){p(x)})/(x-p(x)) is g(p(x))

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limit as x approaches p(x) of (g(x){x}-g(x){p(x)})/(x-p(x))

x \to p (x) lim (\frac{g ( x ) { x } - g ( x ) { p ( x ) }}{x - p ( x )})

g (p (x))

Solution steps

x \to p (x) lim (\frac{g ( x ) { x } - g ( x ) { p ( x ) }}{x - p ( x )})

Plug in the value

x = p (x)

, assuming the function is continuous at this point

= \frac{g ( p ( x ) ) { p ( x ) } - g ( p ( x ) ) { p ( p ( x ) ) }}{p ( x ) - p ( p ( x ) )}

Simplify

\frac{g ( p ( x ) ) ( p ( x ) ) - g ( p ( x ) ) ( p ( p ( x ) ) )}{p ( x ) - p ( p ( x ) )} : g (p (x))

= g (p (x))

What is the limit as x approaches p(x) of (g(x){x}-g(x){p(x)})/(x-p(x)) ?
The limit as x approaches p(x) of (g(x){x}-g(x){p(x)})/(x-p(x)) is g(p(x))