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Final answer:
Limits in calculus are about the value that f(x) approaches as x gets near to a specific point or infinity. To prove a limit exists, we use epsilon and delta to rigorously demonstrate how the function's values can remain close to a limit value. Limits may not exist if no such value can be determined.
Explanation:
The question is about limits in calculus, which are fundamental to understanding the behavior of functions as the inputs approach a particular value. Limits at a point can be one-sided (from the left or the right) or two-sided, and they help to define continuity and derivatives. When we say limx→a f(x), we are asking what value f(x) approaches as x gets closer and closer to a. If there is a number L that f(x) gets arbitrarily close to as x approaches a, then L is the limit. If no such L exists, then the limit does not exist.
To prove a limit exists, we use a formal definition involving two quantities ε (epsilon) and δ (delta). Epsilon represents how close we want f(x) to be to the limit L, and delta represents how close x needs to be to a. This is a rigorous way of showing that for any small distance away from L we choose, we can find a distance away from a such that f(x) will be within that small distance from L.
It is also possible to talk about limits at infinity, which describe the behavior of f(x) as x grows without bound. If f(x) approaches a certain value L as x goes to positive or negative infinity, then L is the limit at infinity of f(x).
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Rewritten by : Barada
SOLUTION:
Case: Limits
Method:
Right and left-sided limits.
The right sided limits is gotten from the right-hand side and gives an f(x) of 0 while the left sided limits gives an f(x) of 1.
Final answer: Option
[tex]\begin{gathered} \lim_{x\to0^-}f(x)=1 \\ \lim_{x\to0^+}f(x)=0 \end{gathered}[/tex]