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The function has a limit L at an input p if f(x) is "close" to L whenever x is "close" to p. In other words, f(x) becomes closer and closer to L as x moves closer and closer to p. To me that sounds like something that might be better described as a 'target'.
Limits in maths are unique real numbers. Let us consider a real-valued function “f” and the real number “c”, the limit is normally defined as \(\lim _{x \rightarrow c} f(x)=L\). It is read as “the limit of f of x, as x approaches c equals L”.
Instead, you should view limits as a way to describe situations (or ask more interesting problems). The derivative is a perfect example of this. If you want to express the idea of "instantaneous rate of change," you are going to use limits to do this.
A one-sided limit only considers values of a function that approaches a value from either above or below. The right-side limit of a function \(f\) as it approaches \(a\) is the limit \[\lim_{x \to a^+} f(x) = L. \] The left-side limit of a function \(f\) is \[\lim_{x \to a^-} f(x) = L. \]
Example: limx→10 x2 = 5. We know perfectly well that 10/2 = 5, but limits can still be used (if we want!)
In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. [1] Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.
Limits let us ask “What if?”. If we can directly observe a function at a value (like x=0, or x growing infinitely), we don’t need a prediction. The limit wonders, “If you can see everything except a single value, what do you think is there?”. When our prediction is consistent and improves the closer we look, we feel confident in it.
