Describe the right end behavior of the function $$ f(x)=\frac{42 x^{7}+3 x^{2}}{18 x^{6}-3 x} $$ by finding $$ \lim _{x \rightarrow \infty} f(x) $$ $$ \lim _{x \rightarrow \infty} f(x)=\square $$ (Simplify your answer.)

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To determine the right end behavior of the function $$ f(x) = \frac{42x^{7} + 3x^{2}}{18x^{6} - 3x} $$, we need to evaluate the limit as $$ x $$ approaches infinity:

$$
\lim_{x \rightarrow \infty} \frac{42x^{7} + 3x^{2}}{18x^{6} - 3x}
$$

**Step 1: Identify the Leading Terms**

For large values of $$ x $$, the highest degree terms dominate the behavior of the function. Thus, we focus on the leading terms in the numerator and the denominator:

- **Numerator:** $$ 42x^{7} $$
- **Denominator:** $$ 18x^{6} $$

**Step 2: Simplify the Expression Using Leading Terms**

$$
\lim_{x \rightarrow \infty} \frac{42x^{7} + 3x^{2}}{18x^{6} - 3x} \approx \lim_{x \rightarrow \infty} \frac{42x^{7}}{18x^{6}} = \lim_{x \rightarrow \infty} \frac{42}{18} x = \lim_{x \rightarrow \infty} \frac{7}{3} x
$$

**Step 3: Evaluate the Limit**

As $$ x $$ approaches infinity, $$ \frac{7}{3} x $$ also approaches infinity. Therefore, the function grows without bound.

$$
\lim_{x \rightarrow \infty} f(x) = \infty
$$

**Conclusion:**

The right end behavior of the function $$ f(x) $$ is that it increases without bound as $$ x $$ approaches infinity.

$$
\lim_{x \rightarrow \infty} f(x) = \infty
$$

**Final Answer:**

$$
\lim_{x \rightarrow \infty} f(x) = \infty
$$
$$ \lim_{x \rightarrow \infty} f(x) = \infty $$

Describe the right end behavior of the function \( f(x)=\frac{42 x^{7}+3 x^{2}}{18 x^{6}-3 x} \) by finding \( \lim _{x \rightarrow \infty} f(x) \) \( \lim _{x \rightarrow \infty} f(x)=\square \) (Simplify your answer.)

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