I really love how you're doing college level math while approaching everything like you're speaking in front of a bunch of 6th graders. Not the kind of patience and calmness everyone has.
I did it by factoring. Ignore the square root, and just focus on whats inside: 4x^14+x^7. Factoring makes it become x^7(4x^7+1). This goes to -inf * -inf, which is just positive infinity. Square rooting a positive infinity still equals infinity. Now, to deal right the -2x^7. It is -2*-inf. This is also + infinity. Infinity + infinity = infinity. This, the entire limit is infinity.
I don't know if I'm right but here is how I did it: You can guarantee that the expression in the square root will be greater than 0 because x^14 grows faster than x^7. So, the result of the square root will be a positive number. As x goes to negative infinity, -2x^7 goes to infinity. Therefore, when x goes to infinity, the whole expression goes to infinity.
You are awesome sir! I'm now very intersted in math after your videos! I can't speak your language, but try my best to understand everything you explain
I know how to solve these limits but you always find some ways i don't know. Love it! Especially the way to solve the irrational function towards the end. I would have used a change of variable: y=(-x) So that we have: Lim(y->+inf)
In the square root it's probably possible to factorise by x^7. This way we can get it to become the sqrt of x^7(4x^7+1) which is plus infinity ,and by adding -2x^7 the result will still be the same mayby.
have my Calculus exam tomorrow and your videos are so consice and easy to understand. And the way youn explain somehow calms my nerves😂.Your videos have come in clutch
To all of you who suggested 0.25 as the limit (like I did): that's indeed the limit when x goes to infinity - but not the limit when x goes to negative infinity, as the problem had it 😉 like my math teacher said: "take time to read the problem text carefully". Which I clearly didn't ...
You can argue that the 14th power is a higher order infinity. Or that the end behavior of the polynomial would tail off at infinity on the right side. Delegating that to the reader, Lim x->-inf sqrt(4x^14+x^7)-2x^7 Adding a constant in the radical isn’t going to make a difference As the sqrt contains a higher number adding some change doesn’t make a difference sqrt(4x^14+x^7+1/16)≈sqrt(4x^14+x^7) lim x->-inf sqrt([2x^7+1/4]^2)-2x^7 |2x^7+1/4|-2x^7 |-inf|-(-inf)=inf
We can solve this without any calculations too. x¹⁴ is +∞ while x⁷ is -∞ BUT x¹⁴ term's magnitude will be way larger than x⁷ term's magnitude, so we can neglect x⁷ term from square root. After neglecting, only square root of 4x¹⁷ - 2x⁷ remains. Square root of 4x¹⁷ is -2x¹⁷ So the result will be -4x¹⁷ After putting x as -∞, we get ∞+∞ situation, so thst will turn out to be +∞ only. So the answer is +∞
Hi, I have an idea.When x approche -Infinity, √(4x^14+x^7) is approximately =√4x^14.Since √4x^14 is positive, so √4x^14=2 (absolute value x )^7, which is positive infinity, and if we -2x^7, it equals to4(absolute value x)^7, which is also positive infinity. I think this way is more simple
I would have solved it like this: let X>0 then I would have divided the radicand by x^14. Then the limit would have been |/(4+1/x^2) - 27x^7 => sqrt(4+0) - 27^-inf > 2+inf => +inf
Hello Prime Newton, I like your channel! But, you got this limit wrong. One can show that the entire expression never exceeds 1/4 by first just writing the expression under the square root as (2x^7 + 1/4)^2 - 1/16, and then bounding the entire expression from above, as follows: √(4x^14 + x^7) - 2x^7 = √((2x^7 + 1/4)^2 - 1/16) - 2x^7 < √((2x^7 + 1/4)^2) - 2x^7 = 2x^7 + 1/4 - 2x^7 = 1/4 You can actually "see" the limit as well from the modified expression: for large X, the square expression (under the square root) dominates and the impact of subtracting 1/16 is completely negligible, that is, for large X: (2x^7 + 1/4)^2 - 1/16 ≈ (2x^7 + 1/4)^2. This already 'hints' the limit is 1/4. This can also be shown by straighforward algebraic manipulation and use of l'Hospital's rule: first pull out 2x^7 from under the square root, factor the resulting expression (with 2x^7 is a common factor), and then 'flip down' the 2x^7 as 1/2x^7 into the denominator. The resulting expressions tends to "0/0" when x->∞ which means you can apply l'Hospital's rule, and get 1/4 as the limit.
I remember a similar comment I read about this limit problem. I am pretty sure my steps are correct and my answer is also correct. The graph of the function also confirms the answer. Infinity could be a little tricky. I have had to take down some of my videos because I played with infinity 😭.
Ah, you're right. I accidentally took the limit as x goes to infinity, and not to 'negative infinity'. I'm pretty sure the others who suggested 0.25 as the limit did the same thing.
I’m afraid, the result is not plus infinity. The value under the square root is little more, than 4. So, the square root of it is a little more, than 2. And the bottom tends to 0 from the left, but not from right!
Too complicated explanation. Instead sqrt(4x^14+x7) -> 2|x|^7 when x -> -infinity. The second part -2x^7 will go towards +2|x|^7 when x -> -infinity because of the sign. Therefore both together will go to +infinity.
He's correct and you're wrong. Use your calculator to find out that the answer of the limit is +inf regardless of the value of x in the -inf. He proved it perfectly that 1 divided by any number approaching to 0 from the right is +inf.
![Limit of absolute value functions [ lim |x^3 - x|/(x^3 - |x|) as x goes to 0 ]](http://i.ytimg.com/vi/nJ9lEolfVpU/mqdefault.jpg)
I really love how you're doing college level math while approaching everything like you're speaking in front of a bunch of 6th graders. Not the kind of patience and calmness everyone has.
Totally agree.
But this is highschool level in my country
I don't like when he does the easiest steps
I studied math at uni and this video is amazing 👍. The calm, clear and concise explanation is nice.
I did it by factoring.
Ignore the square root, and just focus on whats inside: 4x^14+x^7. Factoring makes it become x^7(4x^7+1). This goes to -inf * -inf, which is just positive infinity. Square rooting a positive infinity still equals infinity.
Now, to deal right the -2x^7. It is -2*-inf. This is also + infinity.
Infinity + infinity = infinity. This, the entire limit is infinity.
inf * inf = indeterminate form
@@waltz251 no its not. It's just infinity.
@@hipeplefulYeah
I don't know if I'm right but here is how I did it:
You can guarantee that the expression in the square root will be greater than 0 because x^14 grows faster than x^7. So, the result of the square root will be a positive number.
As x goes to negative infinity, -2x^7 goes to infinity.
Therefore, when x goes to infinity, the whole expression goes to infinity.
You are awesome sir! I'm now very intersted in math after your videos! I can't speak your language, but try my best to understand everything you explain
Very nice problem and brilliant explications. Thanks a lot
شكرا استاذ تعلمت منك الكثير
I know how to solve these limits but you always find some ways i don't know.
Love it!
Especially the way to solve the irrational function towards the end.
I would have used a change of variable:
y=(-x)
So that we have:
Lim(y->+inf)
In the square root it's probably possible to factorise by x^7. This way we can get it to become the sqrt of x^7(4x^7+1) which is plus infinity ,and by adding -2x^7 the result will still be the same mayby.
have my Calculus exam tomorrow and your videos are so consice and easy to understand. And the way youn explain somehow calms my nerves😂.Your videos have come in clutch
Goodluck
Very nice thank you.
despite being a pre-calc student im really good at guessing these limits and its mostly looking at which x is 'bigger' even tho there both inf
To infinity, and beyond . . .
#BackToTheFuture
To all of you who suggested 0.25 as the limit (like I did): that's indeed the limit when x goes to infinity - but not the limit when x goes to negative infinity, as the problem had it 😉 like my math teacher said: "take time to read the problem text carefully". Which I clearly didn't ...
You can argue that the 14th power is a higher order infinity. Or that the end behavior of the polynomial would tail off at infinity on the right side.
Delegating that to the reader,
Lim x->-inf sqrt(4x^14+x^7)-2x^7
Adding a constant in the radical isn’t going to make a difference
As the sqrt contains a higher number adding some change doesn’t make a difference
sqrt(4x^14+x^7+1/16)≈sqrt(4x^14+x^7)
lim x->-inf sqrt([2x^7+1/4]^2)-2x^7
|2x^7+1/4|-2x^7
|-inf|-(-inf)=inf
What background is this?
We can solve this without any calculations too.
x¹⁴ is +∞ while x⁷ is -∞ BUT x¹⁴ term's magnitude will be way larger than x⁷ term's magnitude, so we can neglect x⁷ term from square root.
After neglecting, only square root of 4x¹⁷ - 2x⁷ remains.
Square root of 4x¹⁷ is -2x¹⁷
So the result will be -4x¹⁷
After putting x as -∞, we get ∞+∞ situation, so thst will turn out to be +∞ only.
So the answer is +∞
I always prefer to change limit to -inf to inf , specially when there is square root in the problem
Hi, I have an idea.When x approche -Infinity, √(4x^14+x^7) is approximately =√4x^14.Since √4x^14 is positive, so √4x^14=2 (absolute value x )^7, which is positive infinity, and if we -2x^7, it equals to4(absolute value x)^7, which is also positive infinity. I think this way is more simple
I would have solved it like this: let X>0 then I would have divided the radicand by x^14. Then the limit would have been |/(4+1/x^2) - 27x^7 => sqrt(4+0) - 27^-inf > 2+inf => +inf
Oh man, the limit equals =∞
well i have diveded by x to the power of 7 and result was -4 . any explination
1/4
Hello Prime Newton, I like your channel! But, you got this limit wrong.
One can show that the entire expression never exceeds 1/4 by first just writing the expression under the square root as (2x^7 + 1/4)^2 - 1/16, and then bounding the entire expression from above, as follows:
√(4x^14 + x^7) - 2x^7 = √((2x^7 + 1/4)^2 - 1/16) - 2x^7 < √((2x^7 + 1/4)^2) - 2x^7 = 2x^7 + 1/4 - 2x^7 = 1/4
You can actually "see" the limit as well from the modified expression: for large X, the square expression (under the square root) dominates and the impact of subtracting 1/16 is completely negligible, that is, for large X: (2x^7 + 1/4)^2 - 1/16 ≈ (2x^7 + 1/4)^2. This already 'hints' the limit is 1/4. This can also be shown by straighforward algebraic manipulation and use of l'Hospital's rule: first pull out 2x^7 from under the square root, factor the resulting expression (with 2x^7 is a common factor), and then 'flip down' the 2x^7 as 1/2x^7 into the denominator. The resulting expressions tends to "0/0" when x->∞ which means you can apply l'Hospital's rule, and get 1/4 as the limit.
I remember a similar comment I read about this limit problem. I am pretty sure my steps are correct and my answer is also correct. The graph of the function also confirms the answer. Infinity could be a little tricky. I have had to take down some of my videos because I played with infinity 😭.
Ah, you're right. I accidentally took the limit as x goes to infinity, and not to 'negative infinity'. I'm pretty sure the others who suggested 0.25 as the limit did the same thing.
Yes. I realized it after I responded. And the graph actually shows ¼ for positive infinity.
La réponse est évidente.
+infini-(-infini)=+infini.
I’m afraid, the result is not plus infinity.
The value under the square root is little more, than 4. So, the square root of it is a little more, than 2. And the bottom tends to 0 from the left, but not from right!
I’m sorry, I was wrong. The value in root less than 4, because of odd degree of one of terms.
this is difficult to learn ±∞ and ±√ combo
Who else is 17???
I believe this answer is wrong i checked with both rationalising and a different method
Ok
Too complicated explanation. Instead sqrt(4x^14+x7) -> 2|x|^7 when x -> -infinity. The second part -2x^7 will go towards +2|x|^7 when x -> -infinity because of the sign. Therefore both together will go to +infinity.
This is wrong. The result converges to 0.25. It is incredibly easy to prove this.
Then prove it.
Prove it
He's correct and you're wrong. Use your calculator to find out that the answer of the limit is +inf regardless of the value of x in the -inf. He proved it perfectly that 1 divided by any number approaching to 0 from the right is +inf.
You are wrong. Correct answer is 1/4
Ok
He's correct and you're wrong. Use your calculator to find out that the answer of the limit is +inf