Evaluate the given limit: mathop {lim }limits | vedclass

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Question

(A) Given limit: $\mathop {\lim }\limits_{x 	o 0} \frac{ax + x \cos x}{b \sin x}$ At $x = 0$,the function takes the indeterminate form $\frac{0}{0}$.

Vedclass · Accepted Answer

$\frac{a+1}{b}$

Vedclass · Answer

(A) Given limit: $\mathop {\lim }\limits_{x 	o 0} \frac{ax + x \cos x}{b \sin x}$ At $x = 0$,the function takes the indeterminate form $\frac{0}{0}$. We can rewrite the expression as: $\mathop {\lim }\limits_{x 	o 0} \frac{x(a + \cos x)}{b \sin x} = \frac{1}{b} \mathop {\lim }\limits_{x 	o 0} \left( \frac{x}{\sin x} ight) 	imes \mathop {\lim }\limits_{x 	o 0} (a + \cos x)$ Using the standard limit $\mathop {\lim }\limits_{x 	o 0} \frac{\sin x}{x} = 1$,we have $\mathop {\lim }\limits_{x 	o 0} \frac{x}{\sin x} = 1$. Substituting the values: $= \frac{1}{b} 	imes 1 	imes (a + \cos 0)$ $= \frac{1}{b} 	imes (a + 1)$ $= \frac{a+1}{b}$

Evaluate the given limit: $\mathop {\lim }\limits_{x \to 0} \frac{ax + x \cos x}{b \sin x}$

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