Let's analyze the given limit:
[tex]\[ \lim_{x \to -6^{-}} \frac{x+5}{x+6} \][/tex]
First, let's consider the behavior of the numerator and the denominator as \( x \) approaches -6 from the left side (denoted by the minus sign).
1. Numerator Analysis:
The numerator is \( x + 5 \). As \( x \) approaches -6, \( x + 5 \) approaches:
[tex]\[ -6 + 5 = -1 \][/tex]
So, the numerator \( x + 5 \) approaches -1.
2. Denominator Analysis:
The denominator is \( x + 6 \). As \( x \) approaches -6 from the left, the values of \( x \) are slightly less than -6. Thus, \( x + 6 \) becomes a small negative number approaching 0 from the negative side.
Now let's determine the behavior of the fraction \(\frac{x + 5}{x + 6}\):
- When \( x + 5 \) is approximately -1 (a constant negative value)
- And \( x + 6 \) is a small negative number approaching 0 from the left.
The fraction \(\frac{x + 5}{x + 6}\) can be expressed as:
[tex]\[ \frac{-1}{\text{(a small negative value approaching 0)}} \][/tex]
Since we are dividing -1 by a very small negative number, the result will be positive and very large in magnitude. Therefore, the fraction \(\frac{x + 5}{x + 6}\) approaches negative infinity.
Hence, the limit is:
[tex]\[ \lim_{x \to -6^{-}} \frac{x+5}{x+6} = -\infty \][/tex]
