15 Give a limit expression that describes the left end behavior of the function. \[ f(x)=\frac{30 x^{7}+3 x^{2}}{17 x^{6}-4 x} \]

To analyze the left end behavior of the function \[ f(x) = \frac{30 x^{7} + 3 x^{2}}{17 x^{6} - 4 x}, \] we need to consider the limit of \( f(x) \) as \( x \) approaches negative infinity: \[ \lim_{x \to -\infty} f(x). \] First, we identify the leading terms in the numerator and the denominator: - The leading term in the numerator \( 30 x^7 \). - The leading term in the denominator \( 17 x^6 \). Now, we can simplify the function by focusing on these leading terms: \[ f(x) \approx \frac{30 x^7}{17 x^6} \quad \text{as } x \to -\infty. \] This simplifies to: \[ f(x) \approx \frac{30}{17} x. \] Now, we can compute the limit: \[ \lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} \frac{30}{17} x. \] As \( x \) approaches negative infinity, \( \frac{30}{17} x \) also approaches negative infinity. Therefore, we conclude that: \[ \lim_{x \to -\infty} f(x) = -\infty. \] Thus, the limit expression that describes the left end behavior of the function is: \[ \lim_{x \to -\infty} f(x) = -\infty. \]