Math  /  Calculus

QuestionDetermine the limit shown below in simplest form. limx65x+30x2+8x+12\lim _{x \rightarrow-6} \frac{5 x+30}{x^{2}+8 x+12}

Studdy Solution

STEP 1

What is this asking? We need to find the value that the expression approaches as xx gets really close to 6-6. Watch out! Plugging in x=6x = -6 directly gives us a 00\frac{0}{0} situation, which is undefined!
We'll need some algebraic magic to simplify first.

STEP 2

1. Factor the numerator and denominator
2. Simplify the expression
3. Evaluate the limit

STEP 3

We're factoring the numerator, 5x+305x + 30, to simplify the expression.
We can **factor out a 5**, which gives us 5(x+6)5(x + 6).
So, 5x+30=5(x+6)5x + 30 = 5(x+6).

STEP 4

Now, let's **factor** the denominator, x2+8x+12x^2 + 8x + 12.
We're looking for two numbers that **multiply to 12** and **add up to 8**.
Those numbers are **2 and 6**.
This gives us (x+2)(x+6)(x+2)(x+6).

STEP 5

Now we have 5(x+6)(x+2)(x+6)\frac{5(x+6)}{(x+2)(x+6)}.
Notice that (x+6)(x+6) appears in both the numerator and the denominator!
As long as xx is not exactly 6-6, we can **divide** the numerator and denominator by (x+6)(x+6), which gives us 5x+2\frac{5}{x+2}.
Remember, we're looking at the limit as xx *approaches* 6-6, so it's never actually equal to 6-6.
That means we're safe to simplify!

STEP 6

Now that we've simplified our expression, we can finally **substitute** x=6x = -6 into 5x+2\frac{5}{x+2}.
This gives us 56+2=54\frac{5}{-6+2} = \frac{5}{-4}.

STEP 7

Our **final answer** is 54-\frac{5}{4}.

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