Question

Find the following limit. lim_{xrightarrow+infty}(frac{3-12x}{3x+5}+cscfrac{pi x+pi}{4x})

Limits and continuity
ANSWERED
asked 2021-02-12
Find the following limit.
\(\lim_{x\rightarrow+\infty}(\frac{3-12x}{3x+5}+\csc\frac{\pi x+\pi}{4x})\)

Answers (1)

2021-02-13
Given:
\(\lim_{x\rightarrow+\infty}(\frac{3-12x}{3x+5}+\csc\frac{\pi x+\pi}{4x})\)
Solution:
\(\lim_{x\rightarrow+\infty}(\frac{3-12x}{3x+5}+\csc\frac{\pi x+\pi}{4x})=\lim_{x\rightarrow+\infty}(\frac{3-12x}{3x+5})+\lim_{x\rightarrow\infty}(\csc(\frac{\pi x+\pi}{4x}))\)
\(=\lim_{x\rightarrow+\infty}(\frac{x(\frac{3}{x}-12)}{x(3+\frac{5}{x})})+\lim_{x\rightarrow+\infty}(\csc(\frac{x(\pi+\frac{\pi}{x})}{4x}))\)
\(=\lim_{x\rightarrow+\infty}(\frac{\frac{3}{x}-12}{3+\frac{5}{x}})+\lim_{x\rightarrow+\infty}(\csc(\frac{\pi+\frac{\pi}{x}}{4x}))\)
\(=\frac{0-12}{3+0}+\csc(\frac{\pi+0}{4})\)
\(=-\frac{12}{3}+\csc(\frac{\pi}{4})\)
\(=-4+\sqrt2\)
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