What is the limit as x approaches infinity of csc(1/x)-cot(1/x) ?

The limit as x approaches infinity of csc(1/x)-cot(1/x) is 0

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limit as x approaches infinity of csc(1/x)-cot(1/x)

x \to \infty lim (csc (\frac{1}{x}) - cot (\frac{1}{x}))

0

Solution steps

x \to \infty lim (csc (\frac{1}{x}) - cot (\frac{1}{x}))

Express with sin, cos

= x \to \infty lim (\frac{1 - cos ( \frac{1}{x} )}{sin ( \frac{1}{x} )})

Apply L'Hopital's Rule

: x \to \infty lim \frac{- \frac{s i n ( \frac{1}{x} )}{x ^{2}}}{- \frac{c o s ( \frac{1}{x} )}{x ^{2}}}

= x \to \infty lim \frac{- \frac{s i n ( \frac{1}{x} )}{x ^{2}}}{- \frac{c o s ( \frac{1}{x} )}{x ^{2}}}

Simplify

\frac{- \frac{s i n ( \frac{1}{x} )}{x ^{2}}}{- \frac{c o s ( \frac{1}{x} )}{x ^{2}}} : tan (\frac{1}{x})

= x \to \infty lim (tan (\frac{1}{x}))

Apply the Limit Chain Rule:

0

= 0

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What is the limit as x approaches infinity of csc(1/x)-cot(1/x) ?
The limit as x approaches infinity of csc(1/x)-cot(1/x) is 0