What is the limit as x approaches infinity of (sin(3x))/(sin(4x)) ?

The limit as x approaches infinity of (sin(3x))/(sin(4x)) is diverges

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limit as x approaches infinity of (sin(3x))/(sin(4x))

x \to \infty lim (\frac{sin ( 3 x )}{sin ( 4 x )})

diverges

Solution steps

x \to \infty lim (\frac{sin ( 3 x )}{sin ( 4 x )})

Apply the following algebraic property

: a \cdot b = \frac{a}{\frac{1}{b}}, b \neq = 0

3 x = \frac{x}{\frac{1}{3}}

= x \to \infty lim \frac{sin ( \frac{x}{\frac{1}{3}} )}{sin ( 4 x )}

Simplify

\frac{sin ( \frac{x}{\frac{1}{3}} )}{sin ( 4 x )} : sin (3 x) csc (4 x)

= x \to \infty lim (sin (3 x) csc (4 x))

Apply Limit Divergence Criterion:

diverges

= diverges

x \to \frac{π}{4} lim (7 tan^{t a n (2 x)} (x))

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

x \to a lim (\frac{x + 4}{x - 3})

x \to - 5 lim (17)

x \to - \infty lim ((x)^{3})

What is the limit as x approaches infinity of (sin(3x))/(sin(4x)) ?
The limit as x approaches infinity of (sin(3x))/(sin(4x)) is diverges