Find the limit.limx→3x−3x−3\lim_{x\rarr3}\frac{\sqrt{x}-\sqrt3}{x-3}limx→3x−3x−3

3 6 \frac{\sqrt3}{6} 6 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

Find the limit. lim x → 3 x − 3 x − 3 \lim_{x\rarr3}\frac{\sqrt{x}-\sqrt3}{x-3} lim x → 3 x − 3 x <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> − 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z">

In this problem, we want to find the limit of the square root of x minus the square root of 3 over x minus 3 as x approaches 3. Now here if I were to plug that 3 in directly my denominator would be 0 which I do not want to happen and since I have radicals in this rational multiplying by the conjugate. Now here we're going to take the conjugate of our numerator because that's where our radicals are. So multiplying by the conjugate here the square root of x plus the square root of 3 just reversing that sign in between my two terms. Now since I'm multiplying the numerator I of course need to multiply the denominator by the same thing the square root of x plus the square root of 3 and we can go ahead and fully expand that numerator out multiplying it all together. So I'm still taking the limit as x approaches 3, but fully multiplying that numerator out will end up giving me x minus 3. Now in my denominator, I am not going to fully multiply that out because we'll see what cancels out and I'm going to leave that as x minus 3 times the square root of x plus the square root of 3. Now here I have that common factor x minus 3 that cancels out. And then I am left to find the limit of 1 over the square root of x plus the square root of 3 as x approaches 3 because that's all I'm left with here. So now I can go ahead and plug that 3 in to get my final answer. So I have one over the square root of 3 plus the square root of 3 and combining those square roots on the bottom this ends up being 1 over 2 root 3. But we don't wanna have that radical in the denominator. We wanna go ahead and rationalize here by multiplying by the square root of 3 over the square root of 3. Now in doing that, that ends up getting me the square root of 3 over 6 with that denominator rationalized. So this is my final answer here for the limit of this function as x approaches 3. Let me know if you have any questions here, and I'll see you in the next one.

Find the limit. lim ⁡ x → 3 x − 3 x − 3 \lim_{x\rarr3}\frac{\sqrt{x}-\sqrt3}{x-3} lim x → 3 x − 3 x <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> − 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

3 6 \frac{\sqrt3}{6} 6 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

2 3 2\sqrt3 2 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

1. Limits and Continuity

Finding Limits Algebraically

Business Calculus

1. Limits and Continuity

Finding Limits Algebraically

Struggling with Business Calculus?

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Multiple Choice

Find the limit.
$\lim_{x\rarr3}\frac{\sqrt{x}-\sqrt3}{x-3}$

$2\sqrt3$

$0$

$\frac{\sqrt3}{6}$

Does not exist

Verified step by step guidance

Step 1: Recognize that directly substituting x = 3 into the given limit expression $ \frac{\sqrt{x} - \sqrt{3}}{x - 3} $ results in an indeterminate form $ \frac{0}{0} $. This indicates that we need to simplify the expression further to evaluate the limit.

Step 2: Multiply the numerator and denominator by the conjugate of the numerator, $ \sqrt{x} + \sqrt{3} $, to eliminate the square root in the numerator. This gives: $ \frac{(\sqrt{x} - \sqrt{3})(\sqrt{x} + \sqrt{3})}{(x - 3)(\sqrt{x} + \sqrt{3})} $.

Step 3: Simplify the numerator using the difference of squares formula $ a^2 - b^2 = (a - b)(a + b) $. This results in $ \frac{x - 3}{(x - 3)(\sqrt{x} + \sqrt{3})} $.

Step 4: Cancel the common factor $ x - 3 $ from the numerator and denominator, leaving $ \frac{1}{\sqrt{x} + \sqrt{3}} $. Note that this cancellation is valid because $ x \neq 3 $ in the context of the limit.

Step 5: Substitute $ x = 3 $ into the simplified expression $ \frac{1}{\sqrt{x} + \sqrt{3}} $. This gives $ \frac{1}{\sqrt{3} + \sqrt{3}} = \frac{1}{2\sqrt{3}} $. Simplify further if needed to express the result in the desired form.

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