QuestionFind the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-9}{4 x+12}$ .

Studdy Solution

STEP 1

Assumptions1. We are trying to find the limit as x approaches -3 for the function $\frac{x^{}-9}{4x+12}$ . . We will use the limit properties and algebraic manipulation to simplify the expression.

STEP 2

First, we notice that the numerator is a difference of squares, which can be factored as $(a+b)(a-b)$ where $a=x$ and $b=$ . So, we rewrite the expression as follows $\lim{x \rightarrow-} \frac{(x+)(x-)}{4x+12}$

STEP 3

In the denominator, we can factor out a, which gives us $\lim{x \rightarrow-3} \frac{(x+3)(x-3)}{(x+3)}$

STEP 4

We notice that the term $(x+3)$ is common in both the numerator and the denominator, so we can cancel it out $\lim{x \rightarrow-3} \frac{x-3}{4}$

STEP 5

Now, we can substitute $x = -3$ into the simplified expression $\frac{-3-3}{4}$

STEP 6

Calculate the final value $\frac{-6}{4} = -1.5$ So, $\lim{x \rightarrow-3} \frac{x^{2}-9}{4x+12} = -1.5$ .

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QuestionFind the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-9}{4 x+12}$ .

Studdy Solution

STEP 1

Assumptions1. We are trying to find the limit as x approaches -3 for the function $\frac{x^{}-9}{4x+12}$ . . We will use the limit properties and algebraic manipulation to simplify the expression.

STEP 2

First, we notice that the numerator is a difference of squares, which can be factored as $(a+b)(a-b)$ where $a=x$ and $b=$ . So, we rewrite the expression as follows $\lim{x \rightarrow-} \frac{(x+)(x-)}{4x+12}$

STEP 3

In the denominator, we can factor out a, which gives us $\lim{x \rightarrow-3} \frac{(x+3)(x-3)}{(x+3)}$

STEP 4

We notice that the term $(x+3)$ is common in both the numerator and the denominator, so we can cancel it out $\lim{x \rightarrow-3} \frac{x-3}{4}$

STEP 5

Now, we can substitute $x = -3$ into the simplified expression $\frac{-3-3}{4}$

STEP 6

Calculate the final value $\frac{-6}{4} = -1.5$ So, $\lim{x \rightarrow-3} \frac{x^{2}-9}{4x+12} = -1.5$ .

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QuestionFind the limit: lim⁡x→−3x2−94x+12\lim _{x \rightarrow-3} \frac{x^{2}-9}{4 x+12}limx→−3​4x+12x2−9​.

Studdy Solution

STEP 1

Assumptions1. We are trying to find the limit as x approaches -3 for the function x−94x+12\frac{x^{}-9}{4x+12}4x+12x−9​. . We will use the limit properties and algebraic manipulation to simplify the expression.

STEP 2

First, we notice that the numerator is a difference of squares, which can be factored as (a+b)(a−b)(a+b)(a-b)(a+b)(a−b) where a=xa=xa=x and b=b=b=. So, we rewrite the expression as followslim⁡x→−(x+)(x−)4x+12\lim{x \rightarrow-} \frac{(x+)(x-)}{4x+12}limx→−4x+12(x+)(x−)​

STEP 3

In the denominator, we can factor out a, which gives uslim⁡x→−3(x+3)(x−3)(x+3)\lim{x \rightarrow-3} \frac{(x+3)(x-3)}{(x+3)}limx→−3(x+3)(x+3)(x−3)​

STEP 4

We notice that the term (x+3)(x+3)(x+3) is common in both the numerator and the denominator, so we can cancel it outlim⁡x→−3x−34\lim{x \rightarrow-3} \frac{x-3}{4}limx→−34x−3​

STEP 5

Now, we can substitute x=−3x = -3x=−3 into the simplified expression−3−34\frac{-3-3}{4}4−3−3​

STEP 6

Calculate the final value−64=−1.5\frac{-6}{4} = -1.54−6​=−1.5So, lim⁡x→−3x2−94x+12=−1.5\lim{x \rightarrow-3} \frac{x^{2}-9}{4x+12} = -1.5limx→−34x+12x2−9​=−1.5.

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QuestionFind the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-9}{4 x+12}$ .

Assumptions1. We are trying to find the limit as x approaches -3 for the function $\frac{x^{}-9}{4x+12}$ . . We will use the limit properties and algebraic manipulation to simplify the expression.

First, we notice that the numerator is a difference of squares, which can be factored as $(a+b)(a-b)$ where $a=x$ and $b=$ . So, we rewrite the expression as follows $\lim{x \rightarrow-} \frac{(x+)(x-)}{4x+12}$

In the denominator, we can factor out a, which gives us $\lim{x \rightarrow-3} \frac{(x+3)(x-3)}{(x+3)}$

We notice that the term $(x+3)$ is common in both the numerator and the denominator, so we can cancel it out $\lim{x \rightarrow-3} \frac{x-3}{4}$

Now, we can substitute $x = -3$ into the simplified expression $\frac{-3-3}{4}$

Calculate the final value $\frac{-6}{4} = -1.5$ So, $\lim{x \rightarrow-3} \frac{x^{2}-9}{4x+12} = -1.5$ .