More Than That Limit!

Calculus Level 2

Solve for:

lim ⁡ x → ∞ ( sin ⁡ 10 x π x + 1 ) \displaystyle \lim_{x \to \infty} (\frac {\sin {\frac {10x }{\pi}}}{x} +1)

Write your answer to 2 decimal places.


The answer is 1.

Colin Carmody by Colin Carmody , 21, USA

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1 solution

1 = lim ⁡ x → ∞ − 1 x + 1 ≤ lim ⁡ x → ∞ sin ⁡ 10 x π x + 1 ≤ lim ⁡ x → ∞ 1 x + 1 = 1 ⇒ 1 = \lim_{x\to \infty} \frac{-1}{x} +1 \quad \leq \lim_{x\to \infty} \frac{\sin \frac{10x}{\pi} }{x} +1 \leq \quad \lim_{x\to \infty} \frac{1}{x} +1 = 1 \Rightarrow (apply squeeze theorem or usually named sandwich lemma) lim ⁡ x → ∞ sin ⁡ 10 x π x + 1 = 1 \lim_{x\to \infty} \frac{\sin \frac{10x}{\pi} }{x} +1 = 1

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