lim_(x rarr2)(x-2)/(1-sqrt(3x-5))=

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Identify Limit Form: Identify the form of the limit. We need to determine the form of the limit as x approaches 2. To do this, we substitute x with 2 in the expression (x-2)/(1-sqrt(3x-5)). lim_(x -> 2)(x-2)/(1-sqrt(3x-5)) = (2-2)/(1-sqrt(3*2-5)) = 0/(1-sqrt(6-5)) = 0/(1-sqrt(1)) = 0/0 This is an indeterminate form, which means we need to apply further techniques to evaluate the limit.Apply Conjugate Multiplication: Apply conjugate multiplication to simplify the expression. To resolve the indeterminate form, we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of 1-sqrt(3x-5) is 1+sqrt(3x-5). lim_(x -> 2)(x-2)/(1-sqrt(3x-5)) * (1+sqrt(3x-5))/(1+sqrt(3x-5))Perform Multiplication: Perform the multiplication. Now we multiply the numerators and the denominators. Numerator: (x-2)(1+sqrt(3x-5)) Denominator: (1-sqrt(3x-5))(1+sqrt(3x-5)) The denominator simplifies to 1 - (sqrt(3x-5))^2 by the difference of squares formula.Simplify Denominator: Simplify the denominator. We simplify the denominator using the difference of squares formula. Denominator: 1 - (sqrt(3x-5))^2 = 1 - (3x-5) = 1 - 3x + 5 = 6 - 3xSimplify Numerator: Simplify the numerator. We expand the numerator. Numerator: (x-2)(1+sqrt(3x-5)) = x + x*sqrt(3x-5) - 2 - 2*sqrt(3x-5)Combine Like Terms: Combine like terms in the numerator. Combining like terms, we get: Numerator: x - 2 + x*sqrt(3x-5) - 2*sqrt(3x-5) This simplifies to: (x - 2) + sqrt(3x-5)*(x - 2)Factor Out Common Term: Factor out (x - 2) from the numerator. We factor (x - 2) out of the numerator to get: Numerator: (x - 2)(1 + sqrt(3x-5))Cancel Common Terms: Cancel out the common terms. We now have a common term (x - 2) in both the numerator and the denominator that we can cancel out. lim_(x -> 2)(x - 2)(1 + sqrt(3x-5)) / (6 - 3x) = lim_(x -> 2)(1 + sqrt(3x-5)) / (6 - 3x)Substitute x in Expression: Substitute x with 2 in the simplified expression. Now that we have canceled the common terms, we can substitute x with 2 in the expression to find the limit. lim_(x -> 2)(1 + sqrt(3x-5)) / (6 - 3x) = (1 + sqrt(3*2-5)) / (6 - 3*2) = (1 + sqrt(6-5)) / (6 - 6) = (1 + sqrt(1)) / 0 = (1 + 1) / 0 = 2 / 0

$\lim _{x \rightarrow 2} \frac{x-2}{1-\sqrt{3 x-5}}=$

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lim⁡x→2x−21−3x−5= \lim _{x \rightarrow 2} \frac{x-2}{1-\sqrt{3 x-5}}= limx→2​1−3x−5​x−2​=

$\lim _{x \rightarrow 2} \frac{x-2}{1-\sqrt{3 x-5}}=$