3. Given that lim _(xarrow 0)(f(x))/(Ax^4)+Bx^(2)=C find lim _(xarrow 0)(f(x))/(x^2) 20p

To find the limit of the given expression, we can use the fact that the limit of a quotient is equal to the quotient of the limits, provided that the limit of the denominator is not zero.<br /><br />Given that $\lim _{x\rightarrow 0}\frac {f(x)}{Ax^{4}+Bx^{2}}=C$, we can rewrite this as:<br /><br />$\lim _{x\rightarrow 0}\frac {f(x)}{Ax^{4}} + \lim _{x\rightarrow 0}\frac {f(x)}{Bx^{2}} = C$<br /><br />Since the limit of $f(x)$ as $x$ approaches 0 is 0, the first term becomes 0. Therefore, we have:<br /><br />$\lim _{x\rightarrow 0}\frac {f(x)}{Bx^{2}} = C$<br /><br />Now, we can use the fact that the limit of a quotient is equal to the quotient of the limits:<br /><br />$\lim _{x\rightarrow 0}\frac {f(x)}{Bx^{2}} = \frac{\lim _{x\rightarrow 0} f(x)}{\lim _{x\rightarrow 0} Bx^{2}}$<br /><br />Since the limit of $f(x)$ as $x$ approaches 0 is 0, we have:<br /><br />$\frac{0}{B \cdot 0^{2}} = \frac{0}{0}$<br /><br />This is an indeterminate form, and we need to use L'Hopital's rule to evaluate it. L'Hopital's rule states that if the limit of a quotient is in the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then the limit of the quotient is equal to the limit of the quotient of their derivatives.<br /><br />Differentiating the numerator and denominator with respect to $x$, we get:<br /><br />$\lim _{x\rightarrow 0}\frac {f'(x)}{4Ax^{3}+2Bx}$<br /><br />Evaluating the limit of this expression as $x$ approaches 0, we get:<br /><br />$\frac{f'(0)}{0} = \frac{f'(0)}{0}$<br /><br />This is still an indeterminate form, and we need to use L'Hopital's rule again. Differentiating the numerator and denominator with respect to $x$, we get:<br /><br />$\lim _{x\rightarrow 0}\frac {f''(x)}{12Ax^{2}+2B}$<br /><br />Evaluating the limit of this expression as $x$ approaches 0, we get:<br /><br />$\frac{f''(0)}{2B}$<br /><br />Therefore, the limit of the given expression is $\frac{f''(0)}{2B}$.