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Is There a Difference Between Limit and "two-sided Limit?

**Is there a difference between limit and "two-sided limitâ€?**

It's very much situational take the example $f(x)= left beginarrayll 2 & mboxif $x lt 0$; 1 & mboxif $x ge 0$.endarray

ight.$Here both the left and right limits exist, the left is 2, and the right limit is 1, they are not equal so the limit does not exist.This is because the existence of the left and right limits are a necessary but not a sufficient condition for the limit to exist.I.e existence of the limit $Rightarrow$ left and right limit exist.But the other way does not necessarily hold.

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**Taking out limit from integral of limit.**

Applying the mean value theorem, there exists $xi_y in (y, yh)$ such that$$left|int_a^b fracf(x,yh) - f(x,y)h , dx - int_a^b partial_yf(x,y) , dx

ight| = left|int_a^b partial_y f(x, xi_y) , dx - int_a^b partial_yf(x,y) , dx

ight| leqslant int_a^b |partial_yf(x,xi_y) - partial_yf(x,y)| , dx.$$Since $f_y$ is continuous on $[a,b] times [c,d]$ it is uniformly continuous and for any $epsilon > 0$ there exists $delta >0$ such that if $|h| ightarrow 0 int_a^b fracf(x,yh) - f(x,y)h , dx = int_a^b partial_y f(x,y) , dx = int_a^b lim_h

ightarrow 0 fracf(x,yh) - f(x,y)h , dx . $$.

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**How to find the limit of recursive sequence?**

Let $limlimits_nto inftydfraca_n1a_n=l$. Now, $l=limlimits_nto inftydfraca_n1a_n=1dfrac2l$. Thus $l^2=l2$. The root of this equation gives the limit.

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**Burkeâ€“Schumann limit**

In combustion, Burke-Schumann limit, or large Damkhler number limit, is the limit of infinitely fast chemistry (or in other words, infinite Damkhler number), named after S.P. Burke and T.E.W. Schumann, due to their pioneering work on Burke-Schumann flame. One important conclusion of infinitely fast chemistry is the non-co-existence of fuel and oxidizer simultaneously except in a thin reaction sheet. The inner structure of the reaction sheet is described by Lin's equation

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**Limit state design**

Limit state design (LSD), also known as load and resistance factor design (LRFD), refers to a design method used in structural engineering. A limit state is a condition of a structure beyond which it no longer fulfills the relevant design criteria. The condition may refer to a degree of loading or other actions on the structure, while the criteria refer to structural integrity, fitness for use, durability or other design requirements. A structure designed by LSD is proportioned to sustain all actions likely to occur during its design life, and to remain fit for use, with an appropriate level of reliability for each limit state. Building codes based on LSD implicitly define the appropriate levels of reliability by their prescriptions. The method of limit state design, developed in the USSR and based on research led by Professor N.S. Streletski, was introduced in USSR building regulations in 1955.

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**Convergence, supremum and limit**

A sequence $x_n$ converges to some limit $L$ iff for every $epsilon>0$, there is a $NinmathbbN$ such that if $n>N$, then $|x_n-L|0$, there is some natural number $N$ such that $m,n>N$ implies that $|x_n-x_m|

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**derivative when f(xdx) is hugely different from f(x)**

I am usually not a fan of how derivatives are defined and are explained from an intuition perspective, since the focus is usually on how to take the derivative, rather than why we can take the derivative.Note that the derivative of a single variate function $f(x)$ at $x_0$ is the limit $$limlimits_hto0fracf(xh)-f(x)h$$The existence of this limit roughly implies that there exists some tangent line at $x_0$ to $f(x)$ such that the function will be arbitrarily close to the tangent line depending on how far I zoom in. (I am not usually a fan of thinking about derivatives in terms of tangents, but I think that in this context, it is appropriate). Are there functions where this approximation fails miserably? Most certainly. In fact, most continuous functions are not differentiable (and most functions are not continuous). One of the first continuous functions that was discovered to be nowhere differentiable was the pathological Weierstrass Function.

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