What is the limit as x approaches 0 of (tan(5x))/(7x) ?

The limit as x approaches 0 of (tan(5x))/(7x) is 5/7

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limit as x approaches 0 of (tan(5x))/(7x)

x \to 0 lim (\frac{tan ( 5 x )}{7 x})

\frac{5}{7}

Decimal

0.71428 \dots

Solution steps

x \to 0 lim (\frac{tan ( 5 x )}{7 x})

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

= \frac{1}{7} \cdot x \to 0 lim (\frac{tan ( 5 x )}{x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{sec ^{2} ( 5 x ) \cdot 5}{1})

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

Plug in the value

x = 0

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

Simplify

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

= \frac{5}{7}

x \to - 8 lim (5)

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

x \to 3 lim (\frac{x}{0})

x \to 2 lim (3 x^{2} - 3 x + 11)

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

What is the limit as x approaches 0 of (tan(5x))/(7x) ?
The limit as x approaches 0 of (tan(5x))/(7x) is 5/7