$\infty - \infty = \infty ?$

Calculus Level 2

$\large \lim_{x \to 0} (\csc x - \cot x) = ?$

-\infty

+\infty

1 2

e

-1 0

1 solution

Chew-Seong Cheong
Dec 17, 2017

Relevant wiki: L'Hopital's Rule - Basic

$\begin{aligned} L & = \lim_{x \to o} (\csc x - \cot x) \\ & = \lim_{x \to 0} \left(\frac 1{\sin x} - \frac {\cos x}{\sin x} \right) \\ & = \lim_{x \to 0} \frac {1-\cos x}{\sin x} & \small \color{#3D99F6} \text{A 0/0 cases, the L'Hôpital's rule applies.} \\ & = \lim_{x \to 0} \frac {\sin x}{\cos x} & \small \color{#3D99F6} \text{Differentiate up and down w.r.t. }x. \\ & = \boxed{0} \end{aligned}$

∞ − ∞ = ∞ ? \infty - \infty = \infty ? ∞ − ∞ = ∞ ?

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$\infty - \infty = \infty ?$