"Find limit of limx→0(x6+5x4+7x2cos⁡(1+x−1000)) if it exists" was answered by students like you

Camila Glenn

Answered question

2022-04-13

Find limit of

lim_{x \to 0} (\sqrt{x^{6} + 5 x^{4} + 7 x^{2}} \cos (1 + x^{- 1000}))

if it exists

StettyNagEragpouj

Beginner2022-04-14Added 7 answers

way to prove is using squeeze theorem.
Using

- 1 \leq \cos (x) \leq 1

We've

- \sqrt{x^{6} + 5 x^{4} + 7 x^{2}} \leq {(\sqrt{x^{6} + 5 x^{4} + 7 x^{2}} \cos (1 + x^{- 1000}))} \leq \sqrt{x^{6} + 5 x^{4} + 7 x^{2}}

Thus,

lim_{x \to 0} \sqrt{x^{6} + 5 x^{4} + 7 x^{2}} \leq lim_{x \to 0} {(\sqrt{x^{6} + 5 x^{4} + 7 x^{2}} \cos (1 + x^{- 1000}))} \leq lim_{x \to 0} \sqrt{x^{6} + 5 x^{4} + 7 x^{2}}

\Rightarrow 0 \leq lim_{x \to 0} (\sqrt{x^{6} + 5 x^{4} + 7 x^{2}} \cos (1 + x^{- 1000})) \leq 0

\Rightarrow lim_{x \to 0} (\sqrt{x^{6} + 5 x^{4} + 7 x^{2}} \cos (1 + x^{- 1000})) = 0

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