Find lim_(x rarr1)(2x)/(x^(2)-7x+6). Choose 1 answer: (A) 0 (B) (1)/(14) (C) (1)/(7) (D) The limit doesn't exist

Question

Find  lim_(x rarr1)(2x)/(x^(2)-7x+6). Choose 1 answer: (A) 0 (B)  (1)/(14) (C)  (1)/(7) (D) The limit doesn't exist

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Identify the limit: Identify the limit to be solved. lim_(x -> 1)(2x)/(x^2 - 7x + 6)Check direct substitution: Substitute x = 1 into the denominator to check if the limit can be directly calculated. (1^2 - 7*1 + 6) = 1 - 7 + 6 = 0 Since the denominator equals 0 when x = 1, we cannot directly substitute x = 1 into the expression. We need to simplify the expression further.Factor the quadratic expression: Factor the quadratic expression in the denominator. x^2 - 7x + 6 can be factored into (x - 1)(x - 6).Rewrite the limit: Rewrite the limit using the factored form of the denominator. lim_(x -> 1)(2x)/((x - 1)(x - 6))Cancel out common factor: Cancel out the common factor of (x - 1) from the numerator and the denominator, if possible. We notice that there is no (x - 1) factor in the numerator, so we cannot cancel out any terms. This means we need to find another way to evaluate the limit.Apply L'Hôpital's Rule: Since direct substitution and factor cancellation are not possible, we need to use another method to find the limit. We can use L'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a value c results in an indeterminate form 0/0, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator and then taking the limit of the new function.Differentiate numerator and denominator: Apply L'Hôpital's Rule by differentiating the numerator and the denominator separately. The derivative of the numerator 2x with respect to x is 2. The derivative of the denominator x^2 - 7x + 6 with respect to x is 2x - 7.Rewrite the limit with derivatives: Rewrite the limit using the derivatives. lim_(x -> 1)(2)/(2x - 7)Substitute x = 1: Now, substitute x = 1 into the new expression to find the limit. lim_(x -> 1)(2)/(2*1 - 7) = 2/(2 - 7) = 2/(-5) = -2/5Check for mistakes: Since -2/5 is not one of the given answer choices, we need to check our work for any mistakes. Upon reviewing the steps, we realize that we made a mistake in the previous step. The correct calculation should be: lim_(x -> 1)(2)/(2*1 - 7) = 2/(2 - 7) = 2/(-5) = -2/5 However, -2/5 is not a valid answer because we made a mistake in the sign. The correct calculation is: 2/(2 - 7) = 2/(-5) = -2/5 But since we are looking for positive answers, we must have made an error in our sign during the calculation.

Find $\lim _{x \rightarrow 1} \frac{2 x}{x^{2}-7 x+6}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $0$ $\newline$ (B) $\frac{1}{14}$ $\newline$ (C) $\frac{1}{7}$ $\newline$ (D) The limit doesn't exist

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Find lim⁡x→12xx2−7x+6 \lim _{x \rightarrow 1} \frac{2 x}{x^{2}-7 x+6} limx→1​x2−7x+62x​.\newlineChoose 111 answer:\newline(A) 000\newline(B) 114 \frac{1}{14} 141​\newline(C) 17 \frac{1}{7} 71​\newline(D) The limit doesn't exist

Find $\lim _{x \rightarrow 1} \frac{2 x}{x^{2}-7 x+6}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $0$ $\newline$ (B) $\frac{1}{14}$ $\newline$ (C) $\frac{1}{7}$ $\newline$ (D) The limit doesn't exist