In this episode of Engineer In Training Exam TV, Justin walks you through a Fundamentals of Engineering Exam Review of LHopital Rule.

We will begin by defining L’Hopital’s Rule and run through a couple of instances in which we can use this rule to determine the limit of a particular function.

This Fundamentals of Engineering Exam Review of LHopital Rule is part of the global subject Mathematics.

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Hey what’s going on everyone, it’s Justin Dickmeyer from EngineerInTrainingExam.com.

In today’s video we are going to present a Fundamentals of Engineering Exam Review of LHopital Rule.

We’ll start off this tutorial discussing the theory and then get in to working a problem.

So let’s start off our Fundamentals of Engineering Exam Review of LHopital Rule by defining exactly what LHopital Rule.

let’s

consider the limit as X goes to a of f

of X over G of X if at a both f of X and

G of X are finite and G sub a is not

equal to zero then the typical process

of finding the limit would be plugging

in a wherever we see X in those

functions so our limit would simply be F

of a over G sub a but what happens if

both the numerator and the denominator

10 to 0 it is not clear then what the

limit would be in fact depending on what

functions f of X and G sub X are the

limit can really be anything so when we

get when we approach functions are asked

to find functions and we get an

indeterminate form of say 0/0 that’s not

really telling us much about a limit for

example let me show you a quick example

here let’s take the limit as X goes to 0

of X to the 4th divided by X to the

third so if we are asked to find this

limit we would first try to just plug in

0 for wherever we find X and in this

case doing that would give us the

indeterminate form of 0 over 0 so that’s

not telling us much about what the limit

of that function is so we need to use

l’hopital’s rule to do a couple steps to

help us determine what the real limit of

that function is so let’s take a look at

the first indeterminate form of 0/0 so

suppose that we have that same limit f

of X over G sub X as X goes to a

and when we take the limit in our

standard procedures we find that both f

of X and G of X are going to be 0/0 so

l’hopital’s rule what it does is it

gives us two two rules here number one

if we can take the derivative of both f

of X and G of X then the limit as X goes

to a of f prime of X divided by G prime

of X if that exists and is some number L

then the limit of our original function

f of X over G sub X is also L and number

two it tells us that if in on the other

hand F prime of X divided by G prime of

X if that tends to plus or minus

infinity then our original function also

tends to plus or minus infinity so let’s

take a quick look at an example here

let’s take the limit as X goes to 1 of 2

natural log of X divided by X minus 1 so

what we’d try to do here first is just

plug in 1 wherever we see X and we find

real quickly that we would get the

indeterminate form of 0 over 0 so what

we need to do is employ la PETA’s rule

and the first thing we need to do is

take the derivatives of both the

numerator and the denominator and once

that’s complete all we need to do is

plug that back into the limit and take

the limit so let’s take the limit as X

goes to 1 and take the derivatives of

both the numerator and the denominator

and we get 2 divided by X divided by X

oops sorry divided by 1 so plug in 1

wherever we see X and we find that the

that the limit is going to be 2 and la

PETA’s rule excuse me tells us that the

limit of

the original function is now – so what

if we have the indeterminate form or

another and rather another indeterminate

form we might encounter is infinity over

infinity so this isn’t telling us much

again about the limit so we have to

employ la PETA’s rule again and the same

rules apply if we can find a limit at F

prime of X G prime of X if we can find

this limit then that’s good that’s the

limit of the original function and in

the same way if we find that F prime of

X G prime of X tends to plus or minus

infinity then we know that our original

function also will tend to plus or minus

infinity so let’s take a look at an

example here let’s look at the example

limit as X goes to infinity of e sub X

divided by X so our our first shot here

will just plug in an infinity into

wherever we see X and we’ll see that we

have an indeterminate form of infinity

over infinity so once again not telling

us much about the limit of the function

so what we need to do is take the

derivatives of both the numerator and

the denominator and we will get limit as

X goes to infinity of e sub X over 1 so

now plugging in infinity we see that the

function will tend to plus infinity so

that tells us what our limit of the

original function does sometimes you

know we’ll encounter

equations in which we have to use

l’hopital’s rule of multiple times what

I mean by that is will will be given in

a function say our function is 1 minus

cosine of X divided by x squared and

we’ll be asked to take that limit

and let’s say it approaches zero so if

we plug in zero here we’ll find that

once again we’ll get the indeterminate

form zero over zero so now that we’ve

been through the process we know that we

need to take the derivative of both the

numerator and the denominator using

l’hopital’s rule to determine the limit

so what so let’s do that real quick

let’s take the limit as X goes to zero

of the derivatives and we’ll see that

that’s sine X divided by 2x so usually

right here well we can find a limit

usually it takes one step in using

l’hopital’s rule to find that limit but

in this instance we plug in zero and

once again we get 0/0 so what do you do

in this case it’s it’s not very

difficult all you need to do is employ

la PETA’s rule again take the derivative

of both functions and plug it back into

the limit X goes to 0 of cosine of X

divided by 2 and so now in this case we

plug in 0 and we find that the limit is

1/2 and that’s going to be the limit of

our original function so if you continue

to hit indeterminate forms as you work

your way through la PETA’s rule just

continue to employ until you find that

that definitive limit so that’s it for

now guys I appreciate you guys taking

the time out to check out this quick

tutorial head on over to engineer in

training exam comm for more resources as

you prepare for the engineering training

exam and if you have any questions just

contact me through the site or here on

YouTube

I appreciate have a good day guys but

you

—

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