Questions: Evaluate the following limit: lim as x approaches -∞ of (6x^2 + 17x - 3) / (5x^2 + 14x - 3).

Transcript text: Evaluate the following limit: $\lim _{x \rightarrow-\infty} \frac{6 x^{2}+17 x-3}{5 x^{2}+14 x-3}$.

Solution

Solution Steps

Step 1: Identify the Leading Terms

To evaluate the limit $\lim _{x \rightarrow-\infty} \frac{6 x^{2}+17 x-3}{5 x^{2}+14 x-3}$, we first identify the leading terms in both the numerator and the denominator. The leading term in the numerator is $6x^2$, and the leading term in the denominator is $5x^2$.

Step 2: Simplify the Expression

As $x$ approaches $-\infty$, the lower-degree terms ($17x$, $-3$, $14x$, and $-3$) become insignificant compared to the leading terms. Therefore, the expression simplifies to:

\[ \frac{6x^2}{5x^2} \]

Step 3: Calculate the Limit

The simplified expression $\frac{6x^2}{5x^2}$ can be further reduced to $\frac{6}{5}$ since the $x^2$ terms cancel each other out. Thus, the limit is:

\[ \lim _{x \rightarrow-\infty} \frac{6 x^{2}+17 x-3}{5 x^{2}+14 x-3} = \frac{6}{5} \]