The function f in the figure satisfies limx →2 f(x)=4 ....

The function $f$ in the figure satisfies $\lim _{x \rightarrow 2} f(x)=4 .$ Determine the largest value of $\delta>0$ satisfying each statement.
a. If $0<|x-2|<\delta$ then $|f(x)-4|<1$
b. If $0<|x-2|<\delta$ then $|f(x)-4|<1 / 2$

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Key Concepts

Limit of a Function

This concept refers to the value that a function approaches as the input nears a particular point. Understanding limits allows one to grasp the behavior of functions near points of interest, even if the function is not defined at that exact point.

Epsilon-Delta Definition of a Limit

This is a formal approach to defining limits. For any given positive number ? (epsilon), there exists a corresponding positive number ? (delta) such that if the distance between x and the point of interest is less than ?, then the distance between f(x) and the limit value is less than ?. This definition is fundamental in proving and understanding limits rigorously.

Neighborhoods and Their Role in Limits

A neighborhood around a point refers to an interval that includes all points within a certain distance from that point. In the context of the epsilon-delta definition, the ?-neighborhood of the input ensures that all function values within this interval lie within an ?-distance of the limit, which is critical in assessing function behavior near the point of interest.

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