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Let \( f, g, f_{1}, f_{2}: R \rightarrow R \) be twice differentiable function, and \( f(x) \geq 0, f^{\prime}(x) \geq 0 \), \( g^{\prime}(x)0 \forall x \in R \).
\( \mathrm{P} \)
W
Also \( \lim _{x \rightarrow \infty} f_{1}(x)=5, \lim _{x \rightarrow \infty} f_{2}(x)=12, \lim _{x \rightarrow \infty} f(x)=\infty, \lim _{x \rightarrow \infty} g(x)=\infty \) and \( \frac{f^{\prime}(x)}{g^{\prime}(x)}+f_{1}(x) \frac{f(x)}{g(x)}=f_{2}(x) \forall x0 \).
(Also \( f^{\prime}(x) \) and \( g^{\prime}(x) \) are continuous).
\( \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)} \) equals :
(a) 1
(b) 0
(c) \( \frac{1}{2} \)
(d) 2
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