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

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
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Solution

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Solution Steps

To determine the infinite limit as x x approaches 5-5 from the right, we need to analyze the behavior of the function x+6x+5\frac{x+6}{x+5} as x x gets closer to 5-5 from values greater than 5-5. 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)=x+6x+5 f(x) = \frac{x+6}{x+5} We need to find the limit as x x approaches 5-5 from the right: limx5+x+6x+5 \lim_{x \to -5^+} \frac{x+6}{x+5}

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

As x x approaches 5-5 from the right (x5+ x \to -5^+ ):

  • The numerator x+6 x + 6 approaches 5+6=1 -5 + 6 = 1 .
  • The denominator x+5 x + 5 approaches 5+5=0 -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 x+6x+5\frac{x+6}{x+5} will grow without bound in the positive direction.

Final Answer

\boxed{\infty}

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