\( \lim _{x \rightarrow 0} (\frac{1}{x^{2}}-\cot (x^{2})) \)

La expresión dada no se puede calcular directamente. Sin embargo, podemos simplificarla y luego calcular el límite. La expresión dada es: \( \lim _{x \rightarrow 0} (\frac{1}{x^{2}}-\cot (x^{2})) \) Para simplificarla, recordemos que \( \cot(x) = \frac{1}{\tan(x)} \). Por lo tanto, podemos reescribir la expresión como: \( \lim _{x \rightarrow 0} (\frac{1}{x^{2}}-\frac{1}{\tan (x^{2})}) \) Ahora, podemos calcular el límite de esta expresión simplificada. Evaluate the limit by following steps: - step0: Evaluate using L'Hopital's rule: \(\lim _{x\rightarrow 0}\left(\frac{1}{x^{2}}-\frac{1}{\tan\left(x^{2}\right)}\right)\) - step1: Transform the expression: \(\lim _{x\rightarrow 0}\left(\frac{\tan\left(x^{2}\right)-x^{2}}{x^{2}\tan\left(x^{2}\right)}\right)\) - step2: Use the L'Hopital's rule: \(\lim _{x\rightarrow 0}\left(\frac{\frac{d}{dx}\left(\tan\left(x^{2}\right)-x^{2}\right)}{\frac{d}{dx}\left(x^{2}\tan\left(x^{2}\right)\right)}\right)\) - step3: Find the derivative: \(\lim _{x\rightarrow 0}\left(\frac{2\sec^{2}\left(x^{2}\right)\times x-2x}{2x\tan\left(x^{2}\right)+2x^{3}\sec^{2}\left(x^{2}\right)}\right)\) - step4: Simplify the expression: \(\lim _{x\rightarrow 0}\left(\frac{1-\cos^{2}\left(x^{2}\right)}{\sin\left(x^{2}\right)\cos\left(x^{2}\right)+x^{2}}\right)\) - step5: Use the L'Hopital's rule: \(\lim _{x\rightarrow 0}\left(\frac{\frac{d}{dx}\left(1-\cos^{2}\left(x^{2}\right)\right)}{\frac{d}{dx}\left(\sin\left(x^{2}\right)\cos\left(x^{2}\right)+x^{2}\right)}\right)\) - step6: Find the derivative: \(\lim _{x\rightarrow 0}\left(\frac{4x\cos\left(x^{2}\right)\sin\left(x^{2}\right)}{2x\cos^{2}\left(x^{2}\right)-2x\sin^{2}\left(x^{2}\right)+2x}\right)\) - step7: Rewrite the expression: \(\lim _{x\rightarrow 0}\left(\frac{2\cos\left(x^{2}\right)\sin\left(x^{2}\right)}{\cos^{2}\left(x^{2}\right)-\sin^{2}\left(x^{2}\right)+1}\right)\) - step8: Rewrite the expression: \(\frac{\lim _{x\rightarrow 0}\left(2\cos\left(x^{2}\right)\sin\left(x^{2}\right)\right)}{\lim _{x\rightarrow 0}\left(\cos^{2}\left(x^{2}\right)-\sin^{2}\left(x^{2}\right)+1\right)}\) - step9: Calculate: \(\frac{0}{\lim _{x\rightarrow 0}\left(\cos^{2}\left(x^{2}\right)-\sin^{2}\left(x^{2}\right)+1\right)}\) - step10: Calculate: \(\frac{0}{2}\) - step11: Calculate: \(0\) El límite de la expresión \( \lim _{x \rightarrow 0} (\frac{1}{x^{2}}-\cot (x^{2})) \) es igual a 0.