$$ \left. \begin{array} { l } { f ( x ) _ { 5 } \frac { x } { a x ^ { r } - 3 a x + b } } \ { \lim _ { x ightarrow ( - 1 ) ^ { + } } ^ { f ( x ) = - \infty } \quad \lim _ { x ightarrow 4 ^ { + } } f ( x ) = ? } \end{array} ight. $$

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$$ \left. \begin{array} { l } { f ( x ) _ { 5 } \frac { x } { a x ^ { r } - 3 a x + b } } \ { \lim _ { x ightarrow ( - 1 ) ^ { + } } ^ { f ( x ) = - \infty } \quad \lim _ { x ightarrow 4 ^ { + } } f ( x ) = ? } \end{array} ight. $$

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To solve the problem, we need to analyze the function $$ f(x) = \frac{x}{a x^r - 3 a x + b} $$ and determine the limit as $$ x $$ approaches $$ 4 $$ from the right, given the conditions about the limits at $$ x \rightarrow (-1)^+ $$. ### Step 1: Analyze the limit as $$ x \rightarrow (-1)^+ $$ The condition states that $$ \lim_{x \rightarrow (-1)^+} f(x) = -\infty $$. This implies that the denominator must approach zero from the negative side as $$ x $$ approaches $$-1$$. Therefore, we need to find the values of $$ a $$, $$ b $$, and $$ r $$ such that: $$ a(-1)^r - 3a(-1) + b = 0 $$ This simplifies to: $$ -a + 3a + b = 0 \implies 2a + b = 0 \implies b = -2a $$ ### Step 2: Substitute $$ b $$ into the function Now, substituting $$ b = -2a $$ into the function gives us: $$ f(x) = \frac{x}{a x^r - 3a x - 2a} $$ ### Step 3: Analyze the limit as $$ x \rightarrow 4^+ $$ Next, we need to find $$ \lim_{x \rightarrow 4^+} f(x) $$: $$ f(4) = \frac{4}{a(4^r) - 3a(4) - 2a} $$ This simplifies to: $$ f(4) = \frac{4}{a(4^r - 12 - 2)} $$ $$ f(4) = \frac{4}{a(4^r - 14)} $$ ### Step 4: Determine the behavior of the limit To determine the limit as $$ x $$ approaches $$ 4 $$ from the right, we need to analyze the sign of the denominator $$ a(4^r - 14) $$: 1. If $$ a > 0 $$ and $$ 4^r - 14 > 0 $$, then $$ f(4) $$ approaches a positive value. 2. If $$ a > 0 $$ and $$ 4^r - 14 < 0 $$, then $$ f(4) $$ approaches a negative value. 3. If $$ a < 0 $$ and $$ 4^r - 14 > 0 $$, then $$ f(4) $$ approaches a negative value. 4. If $$ a < 0 $$ and $$ 4^r - 14 < 0 $$, then $$ f(4) $$ approaches a positive value. ### Conclusion To find the specific limit $$ \lim_{x \rightarrow 4^+} f(x) $$, we need to know the values of $$ a $$ and $$ r $$. However, without additional information about $$ a $$ and $$ r $$, we cannot determine the exact limit. If you have specific values for $$ a $$ and $$ r $$, please provide them, and I can calculate the limit accordingly. Cannot determine the limit $$ \lim_{x \rightarrow 4^+} f(x) $$ without knowing the values of $$ a $$ and $$ r $$.

\( \left. \begin{array} { l } { f ( x ) _ { 5 } \frac { x } { a x ^ { r } - 3 a x + b } } \\ { \lim _ { x \rightarrow ( - 1 ) ^ { + } } ^ { f ( x ) = - \infty } \quad \lim _ { x \rightarrow 4 ^ { + } } f ( x ) = ? } \end{array} \right. \)

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