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

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

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

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

\frac{5}{7}

Decimal

0.71428 \dots

Solution steps

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= \frac{5}{7}

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

x \to 1 lim (2 + x)

θ \to 2 π + lim (csc (\frac{θ}{2}))

x \to 1 lim (\frac{3}{1 - x})

x \to 4 - lim (\frac{4 + 8}{2})

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