Find lim_(x rarr1)(x-1)/(sqrt(5x-1)-2). Choose 1 answer: (A) (5)/(2) (B) (4)/(5) (C) -(1)/(5) (D) The limit doesn't exist

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Find  lim_(x rarr1)(x-1)/(sqrt(5x-1)-2). Choose 1 answer: (A)  (5)/(2) (B)  (4)/(5) (C)  -(1)/(5) (D) The limit doesn't exist

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Substitute x = 1: To find the limit of the given function as x approaches 1, we first substitute x = 1 into the function to see if it results in an indeterminate form. lim_(x -> 1)(x-1)/(sqrt(5x-1)-2) = (1-1)/(sqrt(5*1-1)-2) = 0/(sqrt(5-1)-2) = 0/(2-2) = 0/0 Since we get 0/0, which is an indeterminate form, we need to use algebraic manipulation to simplify the expression and resolve the indeterminate form.Multiply by conjugate: To eliminate the square root in the denominator, we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of sqrt(5x-1)-2 is sqrt(5x-1)+2. lim_(x -> 1)(x-1)/(sqrt(5x-1)-2) * (sqrt(5x-1)+2)/(sqrt(5x-1)+2) = [(x-1)(sqrt(5x-1)+2)]/[(sqrt(5x-1)-2)(sqrt(5x-1)+2)]Simplify denominator: Now we simplify the denominator using the difference of squares formula: (a-b)(a+b) = a^2 - b^2. The denominator becomes (sqrt(5x-1))^2 - (2)^2 = (5x-1) - 4 = 5x - 5.Expand numerator: The numerator is expanded by distributing (x-1) to both terms in the conjugate (sqrt(5x-1)+2). The numerator becomes x(sqrt(5x-1)) + 2x - sqrt(5x-1) - 2.Factor out common terms: Now we have the following limit: lim_(x -> 1)[x(sqrt(5x-1)) + 2x - sqrt(5x-1) - 2]/(5x - 5) We can simplify this by factoring out common terms and canceling them out.Cancel out common factor: We notice that the numerator and denominator both have terms that can be factored by x-1. The numerator can be rewritten as (x-1)(sqrt(5x-1)+2) and the denominator as 5(x-1). Now we have: lim_(x -> 1)[(x-1)(sqrt(5x-1)+2)]/[5(x-1)]Simplify expression: Since (x-1) is a common factor in both the numerator and the denominator, we can cancel it out, leaving us with: lim_(x -> 1)(sqrt(5x-1)+2)/5Substitute x = 1: Now we can substitute x = 1 into the simplified expression without getting an indeterminate form: lim_(x -> 1)(sqrt(5x-1)+2)/5 = (sqrt(5*1-1)+2)/5 = (sqrt(5-1)+2)/5 = (sqrt(4)+2)/5 = (2+2)/5 = 4/5Final result: The limit as x approaches 1 of the given function is 4/5, which corresponds to answer choice (B).

Find $\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{5 x-1}-2}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{5}{2}$ $\newline$ (B) $\frac{4}{5}$ $\newline$ (C) $-\frac{1}{5}$ $\newline$ (D) The limit doesn't exist

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Find lim⁡x→1x−15x−1−2 \lim _{x \rightarrow 1} \frac{x-1}{\sqrt{5 x-1}-2} limx→1​5x−1​−2x−1​.\newlineChoose 111 answer:\newline(A) 52 \frac{5}{2} 25​\newline(B) 45 \frac{4}{5} 54​\newline(C) −15 -\frac{1}{5} −51​\newline(D) The limit doesn't exist

Find $\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{5 x-1}-2}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{5}{2}$ $\newline$ (B) $\frac{4}{5}$ $\newline$ (C) $-\frac{1}{5}$ $\newline$ (D) The limit doesn't exist