lim_(x rarr3)(sqrt(2x-5)-1)/(x-3)=

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Identify Function & Point: Identify the function and the point at which we need to find the limit. We have the function f(x) = (sqrt(2x-5)-1)/(x-3) and we need to find the limit as x approaches 3.Direct Substitution Check: Direct substitution to check if the limit can be found this way. Let's substitute x = 3 into the function to see if we get a determinate form. f(3) = (sqrt(2*3-5)-1)/(3-3) = (sqrt(6-5)-1)/(0) = (sqrt(1)-1)/0 = (1-1)/0 = 0/0 We get an indeterminate form 0/0, so we cannot use direct substitution to find the limit.Algebraic Manipulation: Apply algebraic manipulation to simplify the expression. To resolve the indeterminate form, we can multiply the numerator and the denominator by the conjugate of the numerator. The conjugate of sqrt(2x-5)-1 is sqrt(2x-5)+1. Let's multiply the numerator and denominator by this conjugate. f(x) = [(sqrt(2x-5)-1)/(x-3)] * [(sqrt(2x-5)+1)/(sqrt(2x-5)+1)]Multiplication & Simplification: Perform the multiplication in the numerator and simplify. When we multiply the numerators, we get: (sqrt(2x-5)-1)(sqrt(2x-5)+1) = (2x-5) - (1)^2 = 2x - 5 - 1 = 2x - 6 The denominator becomes (x-3)(sqrt(2x-5)+1). So now we have: f(x) = (2x - 6) / [(x-3)(sqrt(2x-5)+1)]Factor Out & Simplify: Factor out common terms and simplify the expression further. Notice that 2x - 6 can be factored as 2(x - 3). f(x) = [2(x - 3)] / [(x-3)(sqrt(2x-5)+1)] Now we can cancel out the (x - 3) term in the numerator and denominator. f(x) = 2 / (sqrt(2x-5)+1)Direct Substitution: Now that the expression is simplified, we can try direct substitution again. Let's substitute x = 3 into the simplified function. f(3) = 2 / (sqrt(2*3-5)+1) = 2 / (sqrt(6-5)+1) = 2 / (sqrt(1)+1) = 2 / (1+1) = 2 / 2 = 1

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

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lim⁡x→32x−5−1x−3= \lim _{x \rightarrow 3} \frac{\sqrt{2 x-5}-1}{x-3}= limx→3​x−32x−5​−1​=

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