What is the limit as t approaches 0 of ((tan(t)sec(2t)))/(3t) ?

The limit as t approaches 0 of ((tan(t)sec(2t)))/(3t) is 1/3

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limit as t approaches 0 of ((tan(t)sec(2t)))/(3t)

t \to 0 lim (\frac{( tan ( t ) sec ( 2 t ) )}{3 t})

\frac{1}{3}

Decimal

0.33333 \dots

Solution steps

t \to 0 lim (\frac{tan ( t ) sec ( 2 t )}{3 t})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= \frac{1}{3} \cdot t \to 0 lim (\frac{tan ( t ) sec ( 2 t )}{t})

Apply L'Hopital's Rule

: t \to 0 lim (\frac{sec ^{2} ( t ) sec ( 2 t ) + sec ( 2 t ) tan ( 2 t ) \cdot 2 tan ( t )}{1})

= \frac{1}{3} \cdot t \to 0 lim (\frac{sec ^{2} ( t ) sec ( 2 t ) + sec ( 2 t ) tan ( 2 t ) \cdot 2 tan ( t )}{1})

Plug in the value

t = 0

= \frac{1}{3} \cdot \frac{sec ^{2} ( 0 ) sec ( 2 \cdot 0 ) + sec ( 2 \cdot 0 ) tan ( 2 \cdot 0 ) \cdot 2 tan ( 0 )}{1}

Simplify

\frac{1}{3} \cdot \frac{sec ^{2} ( 0 ) sec ( 2 \cdot 0 ) + sec ( 2 \cdot 0 ) tan ( 2 \cdot 0 ) \cdot 2 tan ( 0 )}{1} : \frac{1}{3}

= \frac{1}{3}

\frac{\partial}{\partial y} (x e^{x y} - 2 y + 1)

ln (x)

\frac{d}{d x} ((x^{2} + y^{2} + z^{2})^{\frac{1}{2}})

\frac{d}{d x} (6 ln (sec (x) + tan (x)))

cos^{3} (x)

What is the limit as t approaches 0 of ((tan(t)sec(2t)))/(3t) ?
The limit as t approaches 0 of ((tan(t)sec(2t)))/(3t) is 1/3