The function f in the figure satisfies limx →2 f(x)=5 Determine the largest value of δ>0 satisfy

step 1 Hello everyone. So today I'm going to be talking about limits and how we can think about limits

The function f in the figure satisfies limx →2 f(x)=5...

The function $f$ in the figure satisfies $\lim _{x \rightarrow 2} f(x)=5$ Determine the largest value of $\delta>0$ satisfying each statement.
a. If $0<|x-2|<\delta$ then $|f(x)-5|<2$
b. If $0<|x-2|<\delta$ then $|f(x)-5|<1$

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Key Concepts

Limit of a Function

This concept expresses the behavior of a function as the input approaches a particular value. In the epsilon-delta definition, stating that the limit of f(x) as x approaches a equals L means that for every small error bound (epsilon), there exists a range (delta) such that if x is within delta of a (excluding a), then f(x) is within epsilon of L.

Epsilon-Delta Definition of a Limit

This is a formal definition used to rigorously describe what it means for a function to approach a limit. The definition states that for every epsilon > 0 there exists a delta > 0 that ensures that whenever 0 < |x - a| < delta, the difference |f(x) - L| is less than epsilon. It encapsulates the idea of making the function's value as close as desired to the limit by restricting the input values sufficiently near the point of interest.

Neighborhood Concept

In the context of limits, a neighborhood refers to an interval around a point. A delta-neighborhood around the point a (i.e., the interval (a - delta, a + delta)) is used to specify how close x must be to a to guarantee that f(x) is within an epsilon-neighborhood of the limit L. The choice of delta depends on the chosen epsilon and captures the precision needed in approaching the limit.

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