Calculus Limits: Verifying the Solution for (x+∆x)^2 - x^2 / x

  • Thread starter Thread starter kendal12
  • Start date Start date
  • Tags Tags
    Limits
AI Thread Summary
The discussion focuses on verifying the limit of the expression ( (x+∆x)^2 - x^2 ) / x as ∆x approaches 0. The initial calculation led to an incorrect result of 2x + a, but after expanding and simplifying, the correct limit was found to be 2x. Participants clarified that dividing by ∆x instead of x was necessary for the correct approach. The final confirmation highlighted that 2x is also the derivative of x². This reinforces the connection between limits and derivatives in calculus.
kendal12
I just started calculus and we are doing limits. I was doing fine until I ran across this problem:

lim ( (x+∆x)^2 - x^2 ) / x
∆x approaches 0

I substituted delta x for a and ended up with 2x + a

I don't think this is right for some reason, but I think I did everything right, can someone please verify my answer so that I know I am on the right track? thanks!
 
Physics news on Phys.org
Just expand it out. (note that a = \Delta x) (x+a )^{2} - x^{2} = 2ax + a^{2}. So we have \frac{2ax + a^{2}}{x}. Divide through by x. What do you notice when you substitute a = 0.
 
Last edited:
I actually messed up when I typed that problem. It should all be over delta x. I'm so sorry! So I divided through by a and got a + 2x. Then substituted o in for a, so I'm left with 2x. Right?
 
2x, you are right !
Remember this calculation, since you will learn soon that 2x is the derivative of x².
 

Thread 'Struggling to understand the concept of this question (ice cube in water optics question)'

TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...
View full post »
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
View full post »

Similar threads

Help Calculating probability between 2 limits for the ground state
Replies
28
Views
2K
Understanding a math step in solving Schrödinger's equation for single particle in a box
Replies
8
Views
916
Why is the square root of x^2 = |x|?
Replies
14
Views
1K
Can A Vector Have Components Of Say Y or Z in the X-Hat Direction?
Replies
8
Views
905
What is the meaning of the intercept of a particular solution to damped oscillator?
Replies
7
Views
915
Calculating the Variance for X^2: How to Find the Correct Solution
Replies
5
Views
1K
[Intro QM] Verification requested on possible solution for a question
Replies
7
Views
1K
##\lim_{x \to0} \left(\dfrac{1}{\sin^2 x}-\dfrac{1}{x^2}\right)##
Replies
5
Views
2K
Find v_1 for projectile using start position, final position and angle
Replies
10
Views
2K
Morin 3.7 -- Block sliding sideways on an inclined plane
Replies
11
Views
1K

Hot Threads

Recent Insights

Back
Top