Problem 34 Medium Difficulty

(a) Use a graph of
$$ f(x) = \frac{\sqrt{3 + x} - \sqrt{3}}{x} $$
to estimate the value of $ \displaystyle \lim_{x \to 0}f(x) $ to two decimal places.

(b) Use a table of values of $ f(x) $ to estimate the limit to four decimal places.

(c) Use the Limit Laws to find the exact value of the limit.

Answer

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Video Transcript

this problem Number thirty four was this tour Calculus Safe Edition, section two point three Pretty use a graft. Half of X equals this corbett of three plus six minus a square to three wrecks to estimate the value How the limit is X purchase here over this function to do it to test in places simply plant dysfunction and as is foreign and we see that it approach is very close to zero point two nine. So this limit and party we will apart that parks me for it to be a hero. Two nine party. He's a table of values of F Testament limited for distant places. Oh, wait. Use the table to plot this function very close to zero and we see that closer than we got two zero from the left and the right, of course, are we approximated the function value to be point two eight. What happened? Two. Four decimal places and the letter there knew it. Improved estimate party use element lab laws to find the exact value of elements. So where you will really write this as a limit on the next page. Limited expert zero of screwed of three sex Explain over minus the square root of And we will take the next step of rationalizing the numerator. Yeah, by multiplying by square root and Papa's X plus the square root of three to the temple about him, which could thiss three plus sex minus three in the numerator divided by X Time's a quality square root of the quantity three point six, plus a spurt of three we see here at the three minus three, it cancels out to zero, and then we're left with her. One acts in the top, which can cancel with the one X and that there no need her leaving us with no limit his expressions. He around, uh, one over in the square root of the quantity three parsecs plus square root of three. And if we evaluate us at X equals zero, that should give us and exact value one over this herd of three plus a screw to three or one over two times the square root of three. And this is our exact solution. This product are this questioned is equal to approximately on zero point two eight eight seven, which is consistent with our yes, um, it's, um, ists from parts A and B