What is the limit as h approaches 0 of ((sqrt(5(x+h)+4))-sqrt(5x+4))/h ?

The limit as h approaches 0 of ((sqrt(5(x+h)+4))-sqrt(5x+4))/h is 5/(2sqrt(5x+4))

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limit as h approaches 0 of ((sqrt(5(x+h)+4))-sqrt(5x+4))/h

h \to 0 lim \frac{( 5 ( x + h ) + 4 ) - 5 x + 4}{h}

\frac{5}{2 5 x + 4}

Solution steps

h \to 0 lim (\frac{5 ( x + h ) + 4 - 5 x + 4}{h})

Multiply by the conjugate of

5 (x + h) + 4 - 5 x + 4 : \frac{5 h}{5 ( x + h ) + 4 + 5 x + 4}

= h \to 0 lim \frac{\frac{5 h}{5 ( x + h ) + 4 + 5 x + 4}}{h}

Simplify

\frac{\frac{5 h}{5 ( x + h ) + 4 + 5 x + 4}}{h} : \frac{5}{5 ( h + x ) + 4 + 5 x + 4}

= h \to 0 lim (\frac{5}{5 ( h + x ) + 4 + 5 x + 4})

Plug in the value

h = 0

= \frac{5}{5 ( 0 + x ) + 4 + 5 x + 4}

Simplify

= \frac{5}{2 5 x + 4}

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What is the limit as h approaches 0 of ((sqrt(5(x+h)+4))-sqrt(5x+4))/h ?
The limit as h approaches 0 of ((sqrt(5(x+h)+4))-sqrt(5x+4))/h is 5/(2sqrt(5x+4))