What is the limit as t approaches 0 of (sin(3t)+4t)/(tsec(t)) ?

The limit as t approaches 0 of (sin(3t)+4t)/(tsec(t)) is 7

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limit as t approaches 0 of (sin(3t)+4t)/(tsec(t))

t \to 0 lim (\frac{sin ( 3 t ) + 4 t}{t sec ( t )})

7

Solution steps

t \to 0 lim (\frac{sin ( 3 t ) + 4 t}{t sec ( t )})

Apply L'Hopital's Rule

: t \to 0 lim (\frac{cos ( 3 t ) \cdot 3 + 4}{sec ( t ) + t sec ( t ) tan ( t )})

= t \to 0 lim (\frac{cos ( 3 t ) \cdot 3 + 4}{sec ( t ) + t sec ( t ) tan ( t )})

Plug in the value

t = 0

= \frac{cos ( 3 \cdot 0 ) \cdot 3 + 4}{sec ( 0 ) + 0 \cdot sec ( 0 ) tan ( 0 )}

Simplify

\frac{cos ( 3 \cdot 0 ) \cdot 3 + 4}{sec ( 0 ) + 0 \cdot sec ( 0 ) tan ( 0 )} : 7

= 7

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What is the limit as t approaches 0 of (sin(3t)+4t)/(tsec(t)) ?
The limit as t approaches 0 of (sin(3t)+4t)/(tsec(t)) is 7