Find lim_(x rarr4)(x-4)/(3-sqrt(x+5)) Choose 1 answer: (A) -6 (B) -(4)/(3) (C) -(10)/(3) (D) The limit doesn't exist

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Find  lim_(x rarr4)(x-4)/(3-sqrt(x+5)) Choose 1 answer: (A) -6 (B)  -(4)/(3) (C)  -(10)/(3) (D) The limit doesn't exist

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Identify Limit Form: Identify the form of the limit as x approaches 4. We need to check if the limit results in an indeterminate form by substituting x = 4 into the expression (x-4)/(3-sqrt(x+5)). Calculate the numerator and denominator separately: Numerator: x - 4 = 4 - 4 = 0 Denominator: 3 - sqrt(x + 5) = 3 - sqrt(4 + 5) = 3 - sqrt(9) = 3 - 3 = 0 Since both the numerator and denominator are 0, we have an indeterminate form of 0/0.Check Indeterminate Form: Simplify the expression to resolve the indeterminate form. We can use algebraic manipulation to simplify the expression. One common technique is to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of the denominator 3 - sqrt(x + 5) is 3 + sqrt(x + 5). Multiply the original expression by (3 + sqrt(x + 5))/(3 + sqrt(x + 5)): lim_(x rarr4) [(x-4)(3 + sqrt(x + 5))]/[(3 - sqrt(x + 5))(3 + sqrt(x + 5))]Simplify Expression: Perform the multiplication in the numerator and use the difference of squares in the denominator. Numerator: (x - 4)(3 + sqrt(x + 5)) Denominator: (3 - sqrt(x + 5))(3 + sqrt(x + 5)) = 3^2 - (sqrt(x + 5))^2 Now, simplify the denominator: Denominator: 9 - (x + 5) = 9 - x - 5 = 4 - xPerform Multiplication: Notice that the numerator can be simplified further. We can expand the numerator: Numerator: x * 3 + x * sqrt(x + 5) - 4 * 3 - 4 * sqrt(x + 5) Simplify the numerator: Numerator: 3x + x * sqrt(x + 5) - 12 - 4 * sqrt(x + 5)Simplify Numerator: Cancel out the common terms in the numerator and denominator. We have a common term of x - 4 in the numerator and 4 - x in the denominator, which can be simplified to -1 when we reverse the order of the terms in the denominator. So, the expression simplifies to: lim_(x rarr4) [-1(3 + sqrt(x + 5))]/[-1] The -1 in the numerator and denominator cancel out, leaving us with: lim_(x rarr4) (3 + sqrt(x + 5))Cancel Common Terms: Evaluate the limit by substituting x = 4 into the simplified expression. lim_(x rarr4) (3 + sqrt(x + 5)) = 3 + sqrt(4 + 5) = 3 + sqrt(9) = 3 + 3 = 6 However, we must remember that we canceled a negative sign in Step 5, so the actual limit is the negative of this result: Final limit: -6

Find $\lim _{x \rightarrow 4} \frac{x-4}{3-\sqrt{x+5}}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-6$ $\newline$ (B) $-\frac{4}{3}$ $\newline$ (C) $-\frac{10}{3}$ $\newline$ D The limit doesn't exist

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Find lim⁡x→4x−43−x+5 \lim _{x \rightarrow 4} \frac{x-4}{3-\sqrt{x+5}} limx→4​3−x+5​x−4​.\newlineChoose 111 answer:\newline(A) −6-6−6\newline(B) −43 -\frac{4}{3} −34​\newline(C) −103 -\frac{10}{3} −310​\newlineD The limit doesn't exist

Find $\lim _{x \rightarrow 4} \frac{x-4}{3-\sqrt{x+5}}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-6$ $\newline$ (B) $-\frac{4}{3}$ $\newline$ (C) $-\frac{10}{3}$ $\newline$ D The limit doesn't exist