Guess the value of the limit (if it exists) by evaluating...

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

$ \displaystyle \lim_{x \to -3}\frac{x^2 - 3x}{x^2 - 9} $,
$ x $ = -2.5, -2.9, -2.95, -2.99, -2.999, -2.9999, -3.5, -3.1, -3.05, -3.01, -3.001, -3.0001

Key Concepts

One-Sided Limits

One-sided limits look at the behavior of a function as the independent variable approaches a specific point from one side (either from above or below). Consistent one-sided limits from both the left and right indicate that the overall limit exists, ensuring the function behaves similarly when approached from either side.

Limit of a Function

This concept involves determining the value that a function approaches as the input nears a specific point. It is fundamental in calculus and helps describe the behavior of a function even at points where the function might not be defined by direct substitution.

Removable Discontinuity

A removable discontinuity occurs in a function when a factor causing the undefined behavior can be canceled from both the numerator and denominator. Even though the function may not be defined at a certain point, the limit exists as the factors causing the discontinuity can be removed.

Transcript

00:02 This is problem number twenty out stewart eighth edition, section two point two guess the value of the limit if it exists by evaluating the function at the given numbers, correct a six to small places the limit as x approaches negative three of the function of x squared minus three x, divided by the quantity x squared minus nine.

00:26 And the given numbers are the following x values that are listed at the bottom in order to guest limit in order to i'm trying to figure out what does them it might equal.

00:40 We first use dysfunction, which we will call it off and evaluated at all of these given excise.

00:52 So take the first one first x value and plug it into the function and then see what the value is and continue with the following values in the spreadsheet.

01:03 We have done that and we have done the calculations, and we can confirm that when you plug in next is make this two point five you get negative.

01:12 Five.

01:14 What next is negative? two point nine you get eight to twenty nine.

01:17 Next is negative.

01:18 Two point nine five you may get a fifty nine one x a negative two point eight nine.

01:23 You get negative to nine nine when x is negative.

01:26 Two point nine nine nine you get made of two thousand nine hundred ninety nine and when x is negative two point nine nine nine nine you get negative twenty nine thousand nine hundred ninety nine.

01:40 We will be stopping here for now because we sense a patter and we see that the closer that this value gets too negative three, which is what the limit is to find out.

01:55 We see that the function approach is very, very large negative number.

02:01 And since this is the one that is closest, teo negative three, we will just keep our eye on that...

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