Help explaining the concept behind this algebraic method

[terse mirage]

The problem is 4(a), i got the answer k = 4. But the official worked out solution for had some intriguing method

What i did was expand, move the variable and do basic algebra. But i checked the step by step solution and it shows (2)²-k = 0... and then solves to get k=4

Theres a few things i dont understand.First of all, the eqn itself shows a graph eqn where it discontinuity occurs when x = 0.

So my questions are, why is it valid (reasoning and logic) behind '=0' and why can I ignore the denominator?

[fickle dust]

Wdym by expansion?

Can you elaborate how did you expand it?

[terse mirage]

wait

ill take a pic

pardon my shitty handwriting

[slender geyser]

l'hopitals allowed?

[terse mirage]

btw its limit (x ->2)

[terse mirage]

? whats that

[slender geyser]

nvm

[fickle dust]

You don’t need that here

[terse mirage]

im just a young student wanting to at least understand what I'm doing in maths

[fickle dust]

You can’t directly remove the limit in general cases

Since line x=2 does not intersect with the function

[terse mirage]

oh sorry will keep that in mind

[fickle dust]

Next, I’ll explain why it works here

[terse mirage]

alright

[fickle dust]

When you are looking for a point which is supposed to be on a function but is removed for some reason

You can use limit to search for it

[terse mirage]

i understand that

[fickle dust]

The point does not exist

[terse mirage]

i do indeed understand this part of the concept

[fickle dust]

Great

[terse mirage]

when you're putting x=2 you're essentially giving a coordinate that doesnt exist

[fickle dust]

The answer key is telling you that when you remove the certain factor which makes the point disappear, you can get the original value of it

[fickle dust]

yes, but that’s what we are looking for

[terse mirage]

okay, so my question is, theres an algebraic way of doing it that is as follows :
(2)² - k = 0

[fickle dust]

We want to predict the value of it

[terse mirage]

i understand that part

[terse mirage]

this

this is how my teacher showed me

it basically ignores the denominator and just inputs = 0..

before it ignores the denominator it basically is saying the f(x) = 0/0

why is the rationale of doing this i cant understand why does it even work

why its possible to remove the denominator also?

[fickle dust]

This is the process of making “ the factor “ disappear

[fickle dust]

Let me show you can example

[terse mirage]

I understand that, it basically creates another function that isnt equivalent to the original function

but how do one think "lets remove denominator?"

[terse mirage]

for example i understand how factorisation works and conjugation but why remove the denominator for no apparent reason

thats my question

[fickle dust]

Because when you remove it, you won’t get any point that doesn’t exist

Which helps you find the value

[fickle dust]

The result remains the same

[terse mirage]

uh i understand that. What i mean is, let me reword for a second

I understand the main concept. Factorisation works because polymonials blah2 and conjugates also blah. Why does removing denominator in this case doesnt give the wrong value whats the proof or reasoning? I didn't learn or forgot this part of algebra

what kind of algebra let one just removes a denominator for a fraction

[fickle dust]

This is the reasoning

No matter what value you plug in, x-1 will disappear

When you plug in 3
You get 2 * 4 / 2

When you plug in 4
You get 3 * 5 / 3

And so on

It is supposed to be 2 when you plug in 1

Confined to (x-1), the point doesn’t exist

[terse mirage]

yeah but theres a discontinuity at x= 1 so x/=1 by removing the denominator through crossing out the same factors you can create a function where limit for x=1 exists. like how f(x)= x/x isnt equivalent to g(x) = 1 since its true for all values of x EXCEPT 0 for f(x)

this is correct yes?

[fickle dust]

Correct

[terse mirage]

yea but im not asking about the calculus concept

im asking about algebra

[terse mirage]

for this you can cross out the factors to remove the denominator

but suddenly going (2)² - k= 0 is sudden

nothing to cross the denominator

i understand why its removed

but i dont understand why it CAN be removed for this case

[fickle dust]

You’re correct, the only difference between f(x) and g(x) is on x=0

What they want to find is g(x)

That’s the concept of limit

[terse mirage]

^

[terse mirage]

.

[fickle dust]

No, you can’t

algebraically speaking, they are not the same

[terse mirage]

?

do you understand my question

i feel like you don't

[fickle dust]

I feel like you don’t understand your own question…

[terse mirage]

wait wait ill write it down

see what i mean? on the first pic theres two equal factors so you can cross it out

but in second pic is ??

[fickle dust]

Oh, you’re referring to this

[terse mirage]

yes thats why i said its an algebraic question

not a calculus...

[fickle dust]

It’s pretty weird actually, your teacher should have made it clear

[terse mirage]

its basically saying

[terse mirage]

?

[terse mirage]

the step goes like this

[terse mirage]

then thus

this

[fickle dust]

Weird enough

[terse mirage]

there obviously must be a reason for this . an algebraic trick, but why does it work is my question it seems too sudden for me

even the fact its going for = 0/0 confuses me.. its already states the function limit is 4/3 so?? doesnt even know whats going on.. whats the logic

[torpid barn]

Yk the thing is

You don’t even need to know what the limit equals

The fact that it exists tells you the value of k

Because it has to return an indeterminate value when you simply plug in 2

Otherwise it would just be an asymptote

Oh it seems you saw that lol

[torpid barn]

The function doesn’t have to exist at a given point for the limit to return a value

That’s quite literally the whole point of limits

It doesn’t matter what the value of the function is at that point

We just wanna know what value the function approaches as you get closer and closer to that point

And again since the denominator is 0 when you plug in 2 directly for x

The limit can only exist if the denominator also is 0 when you plug in 2

Graphically you can visualize this quite easily

Given some function 1/x

As x approaches 0

The function doesn’t approach any concrete value

[terse mirage]

thanks

[terse mirage]

thx

[sterile heron]

Because if you evaluate the limit, below you are gonna have 3(2)-6 wich equals 0, so the only way for the limit to exist, wich we know it does, is by the top equalling 0 aswell, to have the 0/0 case

[terse mirage]

thx i understand

any reason why the denominator CAN be removed (not why it is removed by rather why Can it be removed)

[torpid barn]

You’re not concerned with the value at the point