lim 1-cosax / 1-cosbx

x-0

\lim_{x \to 0} \frac{1 - \cos a x}{1 - \cos b x}

Here we can write the given limits as;

\lim_{a x \to 0} \frac{1 - \cos a x}{a x} \times a x \times \lim_{a x \to 0} \frac{b x}{1 - \cos b x} \times \frac{1}{b x} = \frac{a x}{b x} \times \lim_{a x \to 0} \frac{1 - \cos a x}{a x} \times \lim_{a x \to 0} \frac{b x}{1 - \cos b x} = 0 . . . . . . . . . . . . . {Since \lim_{x \to 0} \frac{1 - \cos x}{x} = 0}

Therefore,

\lim_{x \to 0} \frac{1 - \cos a x}{1 - \cos b x}

= 0