Find lim_(x rarr-3)(sqrt(4x+28)-4)/(x+3). Choose 1 answer: (A) (1)/(2) (B) 1 (C) 2 (D) The limit doesn't exist

Question

Find  lim_(x rarr-3)(sqrt(4x+28)-4)/(x+3). Choose 1 answer: (A)  (1)/(2) (B) 1 (C) 2 (D) The limit doesn't exist

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Identify the form: Identify the form of the limit. We need to find the limit of the function (sqrt(4x+28)-4)/(x+3) as x approaches -3. Let's first substitute x = -3 into the function to see if the limit can be directly calculated. lim_(x -> -3)(sqrt(4x+28)-4)/(x+3) = (sqrt(4(-3)+28)-4)/((-3)+3) = (sqrt(-12+28)-4)/0 = (sqrt(16)-4)/0 = (4-4)/0 = 0/0 This is an indeterminate form, so we cannot directly calculate the limit. We need to use algebraic manipulation to simplify the expression.Apply algebraic manipulation: Apply algebraic manipulation to simplify the expression. To eliminate the indeterminate form, we can multiply the numerator and the denominator by the conjugate of the numerator. The conjugate of sqrt(4x+28)-4 is sqrt(4x+28)+4. lim_(x -> -3)(sqrt(4x+28)-4)/(x+3) * (sqrt(4x+28)+4)/(sqrt(4x+28)+4)Perform the multiplication: Perform the multiplication. Now, we multiply the numerators and the denominators separately. Numerator: (sqrt(4x+28)-4)(sqrt(4x+28)+4) = (4x+28) - 16 Denominator: (x+3)(sqrt(4x+28)+4)Simplify the expressions: Simplify the resulting expressions. Simplify the numerator: (4x+28) - 16 = 4x + 12 Simplify the denominator: We leave it as is for now: (x+3)(sqrt(4x+28)+4) Now the limit expression looks like this: lim_(x -> -3)(4x + 12)/((x+3)(sqrt(4x+28)+4))Factor out common terms: Factor out common terms. We notice that the numerator has a term 4x + 12 which can be factored as 4(x + 3). lim_(x -> -3)4(x + 3)/((x+3)(sqrt(4x+28)+4)) Now we can cancel out the (x+3) term in the numerator and denominator, as long as x is not equal to -3. lim_(x -> -3)4/((sqrt(4x+28)+4))Evaluate the limit: Evaluate the limit. Now that the expression is simplified, we can substitute x = -3 to find the limit. lim_(x -> -3)4/((sqrt(4x+28)+4)) = 4/((sqrt(4(-3)+28)+4)) = 4/((sqrt(-12+28)+4)) = 4/((sqrt(16)+4)) = 4/(4+4) = 4/8 = 1/2

Find $\lim _{x \rightarrow-3} \frac{\sqrt{4 x+28}-4}{x+3}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{2}$ $\newline$ (B) $1$ $\newline$ (C) $2$ $\newline$ (D) The limit doesn't exist

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Find lim⁡x→−34x+28−4x+3 \lim _{x \rightarrow-3} \frac{\sqrt{4 x+28}-4}{x+3} limx→−3​x+34x+28​−4​.\newlineChoose 111 answer:\newline(A) 12 \frac{1}{2} 21​\newline(B) 111\newline(C) 222\newline(D) The limit doesn't exist

Find $\lim _{x \rightarrow-3} \frac{\sqrt{4 x+28}-4}{x+3}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{2}$ $\newline$ (B) $1$ $\newline$ (C) $2$ $\newline$ (D) The limit doesn't exist