What is the limit as x approaches 0 of (sec(x)-cos(x))/(3x^2) ?

The limit as x approaches 0 of (sec(x)-cos(x))/(3x^2) is 1/3

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limit as x approaches 0 of (sec(x)-cos(x))/(3x^2)

x \to 0 lim (\frac{sec ( x ) - cos ( x )}{3 x ^{2}})

\frac{1}{3}

Decimal

0.33333 \dots

Solution steps

x \to 0 lim (\frac{sec ( x ) - cos ( x )}{3 x ^{2}})

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

= \frac{1}{3} \cdot x \to 0 lim (\frac{sec ( x ) - cos ( x )}{x ^{2}})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{sin ( x ) ( 1 + sec ^{2} ( x ) )}{2 x})

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{cos ( x ) ( 1 + sec ^{2} ( x ) ) + 2 sec ^{2} ( x ) tan ( x ) sin ( x )}{2})

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

Plug in the value

x = 0

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

Simplify

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

= \frac{1}{3}

x \to 2 lim (17.4)

x \to \frac{π}{2} lim (\frac{1}{x} sec (x))

x \to - 1 lim (x^{2} + 3)

x \to 0 lim (e^{x} cos (9 x))

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

What is the limit as x approaches 0 of (sec(x)-cos(x))/(3x^2) ?
The limit as x approaches 0 of (sec(x)-cos(x))/(3x^2) is 1/3