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

Find the following limit.
$\underset{x\to +\mathrm{\infty }}{lim}\left(\frac{3-12x}{3x+5}+\mathrm{csc}\frac{\pi x+\pi }{4x}\right)$
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Neelam Wainwright
Given:
$\underset{x\to +\mathrm{\infty }}{lim}\left(\frac{3-12x}{3x+5}+\mathrm{csc}\frac{\pi x+\pi }{4x}\right)$
Solution:
$\underset{x\to +\mathrm{\infty }}{lim}\left(\frac{3-12x}{3x+5}+\mathrm{csc}\frac{\pi x+\pi }{4x}\right)=\underset{x\to +\mathrm{\infty }}{lim}\left(\frac{3-12x}{3x+5}\right)+\underset{x\to \mathrm{\infty }}{lim}\left(\mathrm{csc}\left(\frac{\pi x+\pi }{4x}\right)\right)$
$=\underset{x\to +\mathrm{\infty }}{lim}\left(\frac{x\left(\frac{3}{x}-12\right)}{x\left(3+\frac{5}{x}\right)}\right)+\underset{x\to +\mathrm{\infty }}{lim}\left(\mathrm{csc}\left(\frac{x\left(\pi +\frac{\pi }{x}\right)}{4x}\right)\right)$
$=\underset{x\to +\mathrm{\infty }}{lim}\left(\frac{\frac{3}{x}-12}{3+\frac{5}{x}}\right)+\underset{x\to +\mathrm{\infty }}{lim}\left(\mathrm{csc}\left(\frac{\pi +\frac{\pi }{x}}{4x}\right)\right)$
$=\frac{0-12}{3+0}+\mathrm{csc}\left(\frac{\pi +0}{4}\right)$
$=-\frac{12}{3}+\mathrm{csc}\left(\frac{\pi }{4}\right)$
$=-4+\sqrt{2}$
Jeffrey Jordon