What is the limit as x approaches 0 of (sqrt(x^2+p^2)-p)/(sqrt(x^2+q(x)^2)-q(x)) ?

The limit as x approaches 0 of (sqrt(x^2+p^2)-p)/(sqrt(x^2+q(x)^2)-q(x)) is -(sqrt(p^2)-p)/(q(0))

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limit as x approaches 0 of (sqrt(x^2+p^2)-p)/(sqrt(x^2+q(x)^2)-q(x))

x \to 0 lim \frac{x ^{2} + p ^{2} - p}{x ^{2} + q ( x ) ( x ) ^{2} - q ( x )}

- \frac{p ^{2} - p}{q ( 0 )}

Solution steps

x \to 0 lim (\frac{x ^{2} + p ^{2} - p}{x ^{2} + q ( x ) x ^{2} - q ( x )})

Plug in the value

x = 0

, assuming the function is continuous at this point

= \frac{0 ^{2} + p ^{2} - p}{0 ^{2} + q ( 0 ) \cdot 0 ^{2} - q ( 0 )}

Simplify

\frac{0 ^{2} + p ^{2} - p}{0 ^{2} + q ( 0 ) \cdot 0 ^{2} - q ( 0 )} : - \frac{p ^{2} - p}{q ( 0 )}

= - \frac{p ^{2} - p}{q ( 0 )}

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What is the limit as x approaches 0 of (sqrt(x^2+p^2)-p)/(sqrt(x^2+q(x)^2)-q(x)) ?
The limit as x approaches 0 of (sqrt(x^2+p^2)-p)/(sqrt(x^2+q(x)^2)-q(x)) is -(sqrt(p^2)-p)/(q(0))