What is the limit as θ approaches 0 of (sin(5θ))/(θ+tan(7θ)) ?

The limit as θ approaches 0 of (sin(5θ))/(θ+tan(7θ)) is 5/8

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limit as θ approaches 0 of (sin(5θ))/(θ+tan(7θ))

θ \to 0 lim (\frac{sin ( 5 θ )}{θ + tan ( 7 θ )})

\frac{5}{8}

Decimal Notation

0.625

Solution steps

θ \to 0 lim (\frac{sin ( 5 θ )}{θ + tan ( 7 θ )})

Apply L'Hopital's Rule

: θ \to 0 lim (\frac{cos ( 5 θ ) \cdot 5}{1 + sec ^{2} ( 7 θ ) \cdot 7})

= θ \to 0 lim (\frac{cos ( 5 θ ) \cdot 5}{1 + sec ^{2} ( 7 θ ) \cdot 7})

Plug in the value

θ = 0

= \frac{cos ( 5 \cdot 0 ) \cdot 5}{1 + sec ^{2} ( 7 \cdot 0 ) \cdot 7}

Simplify

\frac{cos ( 5 \cdot 0 ) \cdot 5}{1 + sec ^{2} ( 7 \cdot 0 ) \cdot 7} : \frac{5}{8}

= \frac{5}{8}

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What is the limit as θ approaches 0 of (sin(5θ))/(θ+tan(7θ)) ?
The limit as θ approaches 0 of (sin(5θ))/(θ+tan(7θ)) is 5/8