Find lim_(x rarr-3)(4x^(2)+12 x)/(x^(2)+4x+3). Choose 1 answer: (A) -3 (B) -12 (c) 6 (D) The limit doesn't exist

Write Function: First, let's write down the function whose limit we need to find as x approaches -3: lim_(x -> -3)(4x^2 + 12x) / (x^2 + 4x + 3).Substitute x = -3: Next, we should try to directly substitute x = -3 into the function to see if we can evaluate the limit: (4(-3)^2 + 12(-3)) / ((-3)^2 + 4(-3) + 3).Perform Substitution and Simplify: Now, let's perform the substitution and simplify: (4(9) + 12(-3)) / (9 - 12 + 3).Factor Numerator and Denominator: Simplifying the numerator and denominator gives us: (36 - 36) / (0). Here we notice that the numerator is 0, but so is the denominator, which means we have an indeterminate form 0/0. This means we cannot directly evaluate the limit by substitution.Cancel Common Factor: Since we have an indeterminate form, we should try to factor the numerator and the denominator to see if any common factors can be canceled out: Numerator: 4x^2 + 12x can be factored as 4x(x + 3). Denominator: x^2 + 4x + 3 can be factored as (x + 1)(x + 3).Evaluate Simplified Function: Now, we can cancel the common factor (x + 3) from the numerator and denominator: (4x(x + 3)) / ((x + 1)(x + 3)) becomes 4x / (x + 1) after canceling (x + 3).Substitute x = -3: We can now try to evaluate the limit of the simplified function as x approaches -3: lim_(x -> -3)(4x / (x + 1)).Simplify Result: Substitute x = -3 into the simplified function: 4(-3) / ((-3) + 1) = -12 / (-2).Simplify the result to find the limit: -12 / (-2) = 6.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find
lim_(x rarr-3)(4x^(2)+12 x)/(x^(2)+4x+3).
Choose 1 answer:
(A) -3
(B) -12
(c) 6
(D) The limit doesn't exist

Find $\lim _{x \rightarrow-3} \frac{4 x^{2}+12 x}{x^{2}+4 x+3}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-3$ $\newline$ (B) $-12$ $\newline$ (C) $6$ $\newline$ (D) The limit doesn't exist

View step-by-step help

Full solution

Q. Find

\lim _{x \rightarrow-3} \frac{4 x^{2}+12 x}{x^{2}+4 x+3}

\newline

Choose

1

answer:

\newline

(A)

-3

\newline

(B)

-12

\newline

(C)

6

\newline

(D) The limit doesn't exist

Write Function: First, let's write down the function whose limit we need to find as $x$ approaches $-3$ : $\lim_{x \to -3}\frac{4x^2 + 12x}{x^2 + 4x + 3}$ .
Substitute $x = -3$ : Next, we should try to directly substitute $x = -3$ into the function to see if we can evaluate the limit: $\frac{4(-3)^2 + 12(-3)}{(-3)^2 + 4(-3) + 3}$ .
Perform Substitution and Simplify: Now, let's perform the substitution and simplify: $(4(9) + 12(-3)) / (9 - 12 + 3)$ .
Factor Numerator and Denominator: Simplifying the numerator and denominator gives us: $\newline$ $(36 - 36) / (0)$ . $\newline$ Here we notice that the numerator is $0$ , but so is the denominator, which means we have an indeterminate form $0/0$ . This means we cannot directly evaluate the limit by substitution.
Cancel Common Factor: Since we have an indeterminate form, we should try to factor the numerator and the denominator to see if any common factors can be canceled out: $\newline$ Numerator: $4x^2 + 12x$ can be factored as $4x(x + 3)$ . $\newline$ Denominator: $x^2 + 4x + 3$ can be factored as $(x + 1)(x + 3)$ .
Evaluate Simplified Function: Now, we can cancel the common factor $(x + 3)$ from the numerator and denominator: $\frac{4x(x + 3)}{(x + 1)(x + 3)}$ becomes $\frac{4x}{x + 1}$ after canceling $(x + 3)$ .
Substitute $x = -3$ : We can now try to evaluate the limit of the simplified function as $x$ approaches $-3$ : $\lim_{x \to -3}\left(\frac{4x}{x + 1}\right).$
Simplify Result: Substitute $x = -3$ into the simplified function: $\frac{4(-3)}{((-3) + 1)} = \frac{-12}{(-2)}$ .
Simplify Result: Substitute $x = -3$ into the simplified function: $\frac{4(-3)}{((-3) + 1)} = \frac{-12}{(-2)}$ .Simplify the result to find the limit: $\frac{-12}{(-2)} = 6$ .