Questions: Determine the infinite limit. lim as x approaches -9 from the left of (x+8)/(x+9)

Solution Steps

To determine the infinite limit as \( x \) approaches -9 from the left for the function \( \frac{x+8}{x+9} \), we need to analyze the behavior of both the numerator and the denominator separately.

1. As \( x \) approaches -9 from the left, the numerator \( x+8 \) approaches -1.
2. As \( x \) approaches -9 from the left, the denominator \( x+9 \) approaches 0 from the negative side.

Since the numerator approaches a constant value (-1) and the denominator approaches 0 from the negative side, the fraction \( \frac{x+8}{x+9} \) will approach positive infinity.

As \( x \) approaches \(-9\) from the left, the numerator \( x + 8 \) approaches: \[ x + 8 \rightarrow -9 + 8 = -1 \]

As \( x \) approaches \(-9\) from the left, the denominator \( x + 9 \) approaches: \[ x + 9 \rightarrow -9 + 9 = 0 \] Since \( x \) is approaching \(-9\) from the left, \( x + 9 \) approaches \( 0 \) from the negative side.

Given that the numerator approaches \(-1\) and the denominator approaches \( 0 \) from the negative side, the fraction: \[ \frac{x + 8}{x + 9} \rightarrow \frac{-1}{0^-} = +\infty \]

Final Answer