Suppose the functions f(x) and g(x) are defined for all x and that limx→0f(x)=12 and limx→0g(x)=2. Find the limits as x→0 of the following functions. f(x)cos⁡xx−1

Given limx→0f(x)=12 and limg(x)=2 we have to find f(x)cos⁡xx−1 we know limx→0f(x)⋅g(x)=[limx→0f(x)][limx→0g(x)] and limx→0f(x)g(x)=limx→0f(x)limx→0g(x) limx→0f(x)cos⁡xx−1=[limx→0f(x)]⋅[limx→0cos⁡xx−1] =12limx→0cos⁡xlimx→0(x−1) =12⋅1−1 =−12

Expert Answer to "Suppose the functions f(x) and g(x) are defined..."

High School
Calculus and Analysis
Calculus 1
Limits and continuity

nitraiddQ

Answered question

2021-01-15

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 & Explanation

gotovub

Skilled2021-01-16Added 98 answers

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}

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