Math Snap

STEP 1

What is this asking?
We're trying to find what happens to the expression $\frac{2x^2 + 14x + 20}{x+5}$ when $x$ gets really close to -5.
Watch out!
If we just plug in $x =$-5, we get a zero in the denominator, which is a big no-no!

STEP 2

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

STEP 3

Let's factor that numerator!
We're looking for two numbers that multiply to $(2 \cdot 20) =$ 40 and add up to 14.
Those magic numbers are 10 and 4!

STEP 4

So, we can rewrite the numerator as:
$$2x^2 + 14x + 20 = 2x^2 + 4x + 10x + 20$$

STEP 5

Now, we can factor by grouping:
$$2x^2 + 4x + 10x + 20 = 2x(x+2) + 10(x+2)$$ $$2x(x+2) + 10(x+2) = (2x+10)(x+2)$$

STEP 6

We can factor out a 2 from the first term:
$$(2x+10)(x+2) = 2(x+5)(x+2)$$ Look at that, an $(x+5)$ appeared!
This is great because it will divide to one with the denominator.

STEP 7

Now, let's put that factored numerator back into the original expression:
$$\frac{2x^2 + 14x + 20}{x+5} = \frac{2(x+5)(x+2)}{x+5}$$

STEP 8

Since we're looking at the limit as $x$ approaches $-5$, $x$ is never actually equal to $-5$.
This means that $(x+5)$ is not zero, so we can divide it to one:
$$\frac{2(x+5)(x+2)}{x+5} = 2(x+2)$$ Much simpler, right?

STEP 9

Now, we can finally evaluate the limit by substituting $x =$-5 into our simplified expression:
$$\lim_{x \rightarrow -5} 2(x+2)$$

STEP 10

Let's plug in $x =$-5:
$$2(-5 + 2) = 2(-3) = -6$$

So, the limit is $-6$!