Proof of Limit: $\varepsilon$-$\delta$ Definition

  • Thread starter Thread starter kidia
  • Start date Start date
  • Tags Tags
    Definition Limit
AI Thread Summary
To prove that lim (x+y)/(x²+y²+1) = 0 as (x,y) approaches (0,0) using the ε-δ definition of limits, one must show that for every ε > 0, there exists a δ > 0 such that if √(x²+y²) < δ, then |(x+y)/(x²+y²+1)| < ε. The expression can be bounded by recognizing that |x+y| ≤ √2(x²+y²) and that x²+y²+1 is always greater than or equal to 1. Therefore, the limit can be simplified to |(x+y)/(x²+y²+1)| ≤ √2δ/(1), which can be made less than ε by choosing an appropriate δ. This confirms that the limit is indeed 0 as (x,y) approaches (0,0).
kidia
Messages
65
Reaction score
0
Please I need help on this one.

Use the ( \varepsilon ,\delta ) definition of limit to prove that

lim (x+y/x^2+y^2+1)=0
(x,y)\rightarrow(0,0)
 
Last edited:
Physics news on Phys.org
Well, what have you done on this so far? Do you know of any similar problems that you do know how to do?

By the way, I'm pretty sure you meant (x+y)/(x²+y²+1), not x+y/x²+y²+1.
 
No Hurkyl,I haven't done this kind of limit help me.And it is

lim (x+y/x^2+y^2+1)=0
(x,y)\rightarrow(0,0)
 
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 »

Thread 'Struggle to understand the concept of the 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 »

Similar threads

Electric field of a finite linear distribution of charge
Replies
18
Views
1K
Electric field created by point charges
Replies
6
Views
812
I Basic Measure Theory: Borel ##\sigma##- algebra
Replies
1
Views
2K
B What is a general definition of a limit?
Replies
1
Views
1K
Understanding the Proper Uncertainty Formula to Use
Replies
4
Views
936
5 charges within a square boundary
Replies
7
Views
596
Find the inductance of an inductor in a circuit
Replies
18
Views
2K
Find v_1 for projectile using start position, final position and angle
Replies
10
Views
2K
Force on a charge at the tip of a hollow, charged cone
Replies
5
Views
4K
2D Fraunhofer-diffraction with infinitely long slits
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top