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Problem 23 Easy Difficulty

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

$ \displaystyle \lim_{x \to 4}\frac{\ln x - \ln 4}{x-4} $

Answer

$$\text { For } f(x)=\frac{\ln x-\ln 4}{x-4}$$
$$\begin{array}{|l|c|l|c|}\hline {}{} {x} & {}{} {f(x)} &{}{} {x} & {}{} {f(x)} \\\hline 3.9 & 0.253178 & 4.1 & 0.246926 \\
3.99 & 0.250313 & 4.01 & 0.249688 \\3.999 & 0.250031 & 4.001 & 0.249969 \\3.9999 & 0.250003 & 4.0001 & 0.249997 \\\hline\end{array}$$
It appears that $\lim _{x \rightarrow 4} f(x)=0.25 .$ The graph confirms that result.

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Video Transcript

Okay, so here we have this function F of X. L N X minus Ellen for over x minus four. And we want what we want to do is use a table of values to estimate the value of this limit and then we're going to confirm that graphically. Okay, so how do we construct a table of values? Well, especially because we want to estimate the value of this limit. Basically what we're gonna do is if F of X looks something like this and we're looking for the limit as X approaches for here. What we want to do is plug in some X values to see what this function looks like as the X values get closer and closer and closer to four. So let's do that. Let's plug in some values. If we plug in X equals two and you should probably do this on your calculator because I doubt many people know Ellen to- Ellen four. So in four. Over Too much as far as -2. Well we get if we plug in x equals -2. Yes. F of X is equal to zero 34 657. Okay. Okay so we have 11 value. Let's plug in a couple more. Okay let's plug in x equals three. If we plug in x equals three We get half of three equals zero 2876. 0.2876. Okay, so let's revisit our graph really quickly. Um So it looks like this goes down if this is The x. value to and this is three looks like it goes from 0.34 to 0.28. So it is dipping down there. Okay Let's just keep going f of 3.9 equals 0.253 five three. Um To and then just keep going because remember as X gets closer and closer to four it means X has to pass through these kinds of X values so then we can find what the function approaches. F. 3.99. If we plug that in for X we get 0.2503. Okay, so Pretty cool. Now let's plug in kind of the other side of four Because we're just trying to estimate the value of the limit. It looks like an X -4 here. As you can see it's on the denominator which means we might have to deal with something undefined or something like that. So we just have to approach it from the left and from the right, so this is from the right hand side now the right. If we plug in 4.01 we get Put that into a calculator. You get zero 2497 .2497. Okay, Let's just keep going f of 4.1 is zero point 246 2470 0.24 70. Okay. Just keep going here. You can plug in one more value F. of 4.5 Half of 4.5. When we do that we get 0.23 56. 0.2356. Okay, So now if you put all these into a table with this being the first row, the X. Value and f of X being the second row. Then we'll see that as X approaches for here. As we get closer and closer to four, it looks like this limit approaches 0.25 0.25. All these numbers head down towards 0.25 and from the right hand side it looks like it heads up towards 0.25 up. So and now if you plug this into your calculator to verify that if you have a graphing device, you'll see that Yes. The function looks actually just like this one up here. And when we look at the value x equals four. We see that the function approaches it from the left and the right. So the limit as X approaches four of L n x. penicillin forever, X minus four is about 0.25. Okay, so this is how you estimate it using a table of values. You plug in a bunch of numbers that get really close to this number here. X approaches for which we're looking for. We approach it from the left And from the right we evaluate FFX at all those different places and we see what number it's approaching and what you'll see here. Is that all these numbers are approaching 0.25 From the left, and all these numbers go up to 0.25 from the right, Okay.