Questions: Determine the infinite limit. lim as x approaches -5 from the right of (x+6)/(x+5) ∞ -∞

Transcript text: Determine the infinite limit. \[ \lim _{x \rightarrow-5^{+}} \frac{x+6}{x+5} \] $\infty$ $-\infty$ Submit Answer

Solution

Solution Steps

To determine the infinite limit as $ x $ approaches $-5$ from the right, we need to analyze the behavior of the function $\frac{x+6}{x+5}$ as $ x $ gets closer to $-5$ from values greater than $-5$. Specifically, we should look at the numerator and the denominator separately to understand how the function behaves near this point.

Step 1: Define the Function and the Limit

We are given the function: \[ f(x) = \frac{x+6}{x+5} \] We need to find the limit as $ x $ approaches $-5$ from the right: \[ \lim_{x \to -5^+} \frac{x+6}{x+5} \]

Step 2: Analyze the Behavior of the Function Near $ x = -5 $

As $ x $ approaches $-5$ from the right ($ x \to -5^+ $):

The numerator $ x + 6 $ approaches $ -5 + 6 = 1 $.
The denominator $ x + 5 $ approaches $ -5 + 5 = 0 $ from the positive side.

Step 3: Determine the Sign of the Limit

Since the numerator approaches a positive value (1) and the denominator approaches zero from the positive side, the fraction $\frac{x+6}{x+5}$ will grow without bound in the positive direction.

Final Answer

$\boxed{\infty}$

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