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

Use a table of values to estimate the value of the limit. If...

Key Concepts

Numerical and Graphical Estimation Methods

Using tables of values or graphing tools to estimate limits provides a practical means to approximate the behavior of functions near a point. These methods are especially useful when analytical techniques are complex or when a visual confirmation of the limit is desired.

Limit

Limits are a foundational concept in calculus that describe the value a function approaches as the input nears a specified point. They are essential for understanding the behavior of functions near points of interest, especially when direct substitution yields indeterminate forms.

Derivative as the Limit of a Difference Quotient

The derivative of a function at a particular point is defined as the limit of the difference quotient as the change in the input approaches zero. This concept captures the idea of an instantaneous rate of change, forming the cornerstone of differential calculus.

Natural Logarithm and Its Derivative

The natural logarithm is a fundamental transcendental function, and its derivative is given by 1/x. This result, typically derived from the definition of the derivative, is important for understanding growth processes and appears frequently in both pure and applied mathematics.

Transcript

00:01 Okay, so here we have this function, f of x, lnx minus ln4 over x minus 4, and 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.

00:15 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 going to do is if f of x looks something like this, and we're looking for the limit as x approaches four here.

00:36 What we want to do is plug in us some x values to see what this function looks like as the x values get closer and closer and closer to four.

00:46 So let's do that.

00:48 Let's plug in some values.

00:50 If we plug in x equals 2, and you should probably do this on your calculator, because i doubt many people know ln2 minus ln4.

01:01 Ln4 over 2 minus 4 is negative 2.

01:08 What we get, if we plug in, x equals negative 2, is f of x is equal to 0 .34657.

01:23 Okay, so we have one value.

01:29 Let's plug in a couple more.

01:32 Okay, let's plug in x equals 3.

01:35 If we plug in x equals 3, we get f of 3 equals 0 .2876.

01:47 0 .2876.

01:50 Okay, so let's revisit our graph really quickly.

01:56 So it looks like this goes down, if this is the x value 2, and this is 3, it looks like it goes from 0 .34 to 0 .28.

02:05 So it is dipping down there.

02:07 Okay, let's just keep going.

02:10 F of 3 .9 equals 0 .253.

02:22 532 um 2.

02:29 And then just keep going because remember as x gets closer and closer to 4, it means x has to pass through these kinds of x values.

02:39 So then we can find what the function approaches.

02:43 F of 3 .99, if we plug that in for x, we get 0 .2503.

02:51 Okay, so pretty cool.

02:56 Now let's plug in kind of the other side of 4.

02:59 Because we're just trying to estimate the value of the limit.

03:02 It looks like at x minus 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.

03:12 So we just have to approach it.

03:14 It from the left and from the right...

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