Examples of Limits of Rational Numbers (with Explanation) Video Lecture | Mathematics for Grade 11

hello friends this video limits and
derivatives part 8 is brought to an exam
calm no more fear for exam before
watching this video please make sure
that you have watched part 1 and part 7
normal study limits of a rational
function let there be a rational
function FX is equal to GX by HX from
where H X is not equal to zero obviously
it will be in fine so we want to find
the limit of FX so FX is nothing but G X
by H X so we can say this is nothing but
limit of GX by H X also we know that
limit of GX by HX is equal to limit of
GX by limit of HX this we have learnt in
the algebra of limits and we also know
that limit of GX limit of GX where X
tends to a is nothing but G of a so
that's why we have written this GOP and
similarly limit of HX where X tends is H
so this we can say that limit of the
rational function where extension is
nothing but G a by a chain where FX is
sleek's by H X also we have to observe
here this the G energy comes out to be 0
comma 0 form then we can rewrite the
fact and cancel the factor which makes
it 0 myself if you put the value of x go
to a here in G n H a and you get a value
0 y 0 that means we can rewrite the
factor by canceling the factor which
make it 0 comma 0 if the value comes out
to be K by 0 form then it is undefined
example let's suppose we have this
function X square minus 4 right by X
minus 2 this is the function and here a
is equal to its opposed to now if I put
a G a by a chain that will become 2
squared minus 4 by 2 minus T M 0 by 0 so
that is 0 Y with 0 form then we can
rewrite the factor so what I'll do is X
square minus 4 by X minus 2 I can write
this as form X plus 2 into X minus 2 I
can
experience for in this second 8x with us
too now since we have to cancel the
forum wishmaker zero by zero equal X
minus two is the culprit
which makes this zero because they say -
2 minus 2 is 0 and 2 minus 2 0 so we
will cancer this guy's so what we get is
X plus 2 correct
in case of X plus 2 will not work
excellent - so they'll get 2 plus 2 this
so the limit of this function at X is
going to 2 is nothing but 4 so what we
have done first we have written this
into G a by a chicken and we got 0 by 0
and then we found that since this is 0
by 0 we have refracted this by
cancelling the factors which made it 0
and then we got X plus 2 X Lee here and
in this expect to when we say it's going
to 2 here we get 2 plus 2 is going to
for taking example so let's suppose we
have this equation X cube minus 2 by u
will have its opposed by X minus 2
correct area so we have to find this
limit with respect to X tilde
so here else we will input G 2 by H 2
this is Q minus 2 by 2 minus that is 8
minus 2 by 6 by 0 this is K by 0 and
that means value or limit of this
function at X equal 2 is undefined
correct what we have done here we have
just formed put the value of G 2 by H 2
and we found this value gave out to be 6
by 0 that is our form key by 0 we can
say that the limit of this function at X
cube is undefined make some examples so
here we have to find the value of x cube
by 5 by X plus 20 where X is tending to
1 so what we'll do here here we'll write
G 1 by H 1 first correct that will come
out to be 1 plus 5 by 1
Plus that is six by this is a good one
so they say that limit of this is equal
to 6 by 20
that's ass because had this will be 0 by
0 form you'd have to refactor had this
been ky0 would have told that this is
undefined but this value is K 1 by K 2
correct so that means this is a pile
number and we can say that for XQ plus 5
by X plus 20 you live in x times 1
there's nothing but since pipe maybe one
will take in one sample here also we
have to find this value limit so then
say that this is nothing but G of 2 by H
of 2 that is nothing but G of 2 is 2 2
minus 4 into Q square plus 4 into T by 4
2 square minus R right the derivative of
her to be 0 and the limiter is 8 minus 2
3 4 4 into 4 16 plus P that is 0 by 0
since we have 0 by 0 form that means we
can we have a option to factor it so
let's factor this way so what we get is
X into X square minus 4x plus 4 by this
is X plus 2 into X minus 2 you have
other option to refactor this also this
becomes X into this is nothing but X
plus 2 whole square now X minus 2 whole
square is X minus 2 into X minus 2 by X
plus 2 X minus so will cancel this X
minus factor now this is the guy which
is making X tends to 0 to 0 because 2
minus 2 is 0 2 minus 2 0 will cancel
this factor so what we get here is now
if you put G 2 by H 2 you get is our to
do in T 2 minus T I
that this is you go back in there Israel
so the van tsarism so limit for this
function exists X aluminum and the value
is zero what we have done very simple we
have form Z 2 by H 2 and B format this
value was 0 by 0 form so we refactored
this and this and we have canceled X
minus 2x minus 2 because this was the
culprit it was making this whole
equation 0 by 0 and we have got G 2 by H
2 miles on it signal noise ample here we
have to find limit X tends to be 4 X
square minus 4 by x2 minus 4x plus 4x
here also it's fine
G 2 by H to write this value is 2 square
minus R to the power 4 this is X to the
power 4 sorry Wilbur 4 - phone by 2 to
the power Q minus 4 into square plus 4
into but I have done I just put the
value of x is going to here so what I
get it to the power 4 is equal to 16
minus 4 by 2 or 3 is 8 minus 4 into 4
plus so what you get is 2 L by itself
since we got to n by 0 form that this
undefined correct you know that if I get
something in K by 0 form that means this
limit at X equal to 4 this function is
not defined very simple what we have
done we just formed G 2 by H 2 because
here a was 2 and we got this value let
clear legible and we told that this
value is not defined try out my example
here we have to find again at X equal to
2 so for this also we have to say it G 2
by H 2 and this color comes out to be Q
minus 2 equal to square I'm square minus
Omega T plus 6 so that value if you see
it is a minus to check for a 8 by 4
minus 10 plus the 0 by 0 since it is 0
by 0 form
please we can refactor it so let's
refactor this this becomes X in 2 or X
square will be common this becomes X
square into X minus 2 correct and this
becomes the refactor list this becomes X
square minus 3x minus 2 X plus 6 or we
get X square into X minus 2 here and we
get here X minus 3x is 3x minus 2x minus
2x minus will cancel because this is the
guy which is in culprit because when we
say X going to 2 this becomes 0 so we
can serve this so this becomes x squared
by X minus 3
now in this equation then put X is equal
to 2 so what we get is GT by h2 again
missing this becomes 2 squared by minus
that is over by minus 1 is minus so
minus 4 is the answer is very simple
what we have done B edges on GT yhp form
this value as 0 by 0 form that means we
have a provision to refactor it and we
refactored this guy and we have canceled
this X minus 2 with the next minutes to
close the component that was making it 0
by 0 form and then we have got this
answer as minus here thank you
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