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

The limit as x approaches 0 of (sqrt(x+4)-2)/(sin(5x)) is 1/20

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limit as x approaches 0 of (sqrt(x+4)-2)/(sin(5x))

x \to 0 lim (\frac{x + 4 - 2}{sin ( 5 x )})

\frac{1}{20}

Decimal Notation

0.05

Solution steps

x \to 0 lim (\frac{x + 4 - 2}{sin ( 5 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{\frac{1}{2 x + 4}}{cos ( 5 x ) \cdot 5})

= x \to 0 lim (\frac{\frac{1}{2 x + 4}}{cos ( 5 x ) \cdot 5})

Plug in the value

x = 0

= \frac{\frac{1}{2 0 + 4}}{cos ( 5 \cdot 0 ) \cdot 5}

Simplify

\frac{\frac{1}{2 0 + 4}}{cos ( 5 \cdot 0 ) \cdot 5} : \frac{1}{20}

= \frac{1}{20}

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x \to 0 lim (\frac{1}{x ^{2} - x})

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What is the limit as x approaches 0 of (sqrt(x+4)-2)/(sin(5x)) ?
The limit as x approaches 0 of (sqrt(x+4)-2)/(sin(5x)) is 1/20