Check the solution to "Prove that lim x tends to 0x4cos⁡2x=0"

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2022-08-20

Prove that lim x tends to

0 x^{4} \frac{\cos 2}{x} = 0

g5riem7z

Beginner2022-08-21Added 12 answers

Squeeze theorem
If

h (x) \leq f (x) \leq g (x)

And

lim_{x \Rightarrow a} h (x) = L

And

lim_{x \Rightarrow a} g (x) = L

Then

lim_{x \Rightarrow a} f (x) = L

- 1 \leq \cos (\frac{2}{x}) \leq 1

Multiply Throughout by

x^{4}

, To get

- x^{4} \leq x^{4} \cos (\frac{2}{x}) \leq x^{4}

Since,

lim_{x \Rightarrow 0} - x^{4} = 0

And,

lim_{x \Rightarrow 0} x^{4} = 0

By Squeeze Theorem, We have

lim_{x \Rightarrow 0} x^{4} \cos (\frac{2}{x}) = 0

Result:

lim_{x \Rightarrow 0} x^{4} \cos (\frac{2}{x}) = 0

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