Suppose the functions f(x) and g(x) are defined for all x and that lim_{xrightarrow0}f(x)=frac{1}{2} and lim_{xrightarrow0}g(x)=sqrt2. Find the limits as xrightarrow0 of the following functions. f(x)frac{cos x}{x-1}

Question

Suppose the functions f(x) and g(x) are defined for all x and that

lim_{x \to 0} f (x) = \frac{1}{2}

and

lim_{x \to 0} g (x) = \sqrt{2}

. Find the limits as

x \to 0

of the following functions.

f (x) \frac{\cos x}{x - 1}

Answer 1

Given

lim_{x \to 0} f (x) = \frac{1}{2}

and

lim g (x) = \sqrt{2}

we have to find

f (x) \frac{\cos x}{x - 1}

we know

lim_{x \to 0} f (x) \cdot g (x) = [lim_{x \to 0} f (x)] [lim_{x \to 0} g (x)]

and

lim_{x \to 0} \frac{f (x)}{g (x)} = lim_{x \to 0} \frac{f (x)}{lim_{x \to 0} g (x)}

lim_{x \to 0} \frac{f (x) \cos x}{x - 1} = [lim_{x \to 0} f (x)] \cdot [lim_{x \to 0} \frac{\cos x}{x - 1}]

= \frac{\frac{1}{2} lim_{x \to 0} \cos x}{lim_{x \to 0} (x - 1)}

= \frac{1}{2} \cdot \frac{1}{- 1}

= - \frac{1}{2}

Expert Answer