alguem sabe calcular esse limite? ↩ Lim x->0 xcotg(ax)

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\(\Longrightarrow \underset{x \to 0} \lim \,\, x \cdot \cot(ax) = \underset{x \to 0} \lim \,\, x \cdot {\cos(ax) \over \sin (ax) }\)

\(\Longrightarrow \underset{x \to 0} \lim \,\, x \cdot \cot(ax) = \underset{x \to 0} \lim \,\, \cos(ax) \cdot {x \over \sin (ax) }\)

\(\Longrightarrow \underset{x \to 0} \lim \,\, x \cdot \cot(ax) ={1 \over a} \underset{x \to 0} \lim \,\, \cos(ax) \cdot \Big [ {ax \over \sin (ax) } \Big ] \)

Sabe-se que \(\underset{x \to 0 } \lim { \sin x \over x} =1\). Então, o resultado final é:

\(\Longrightarrow \underset{x \to 0} \lim \,\, x \cdot \cot(ax) ={1 \over a} \cos(a\cdot 0) \cdot [ 1 ] \)

\(\Longrightarrow \underset{x \to 0} \lim \,\, x \cdot \cot(ax) ={1 \over a} \cdot 1 \cdot 1\)

\(\Longrightarrow \fbox {$ \underset{x \to 0} \lim \,\, x \cdot \cot(ax) ={1 \over a} $}\)