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.)

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:** \[ \lim_{x \rightarrow \infty} f(x) = \infty \]