Math / Calculus

QuestionFind the limit $L$ . $L=\lim _{x \rightarrow-5} \frac{2 x^{2}+14 x+20}{x+5}$

Studdy Solution

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: $2(-5 + 2) = 2(-3) = -6$

STEP 11

So, the limit is $-6$ !

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