(a) Prove that if limf(x)=l, then lim|f|(x)=|l| (b) Prove...

(a) Prove that if $\lim f(x)=l,$ then $\lim |f|(x)=|l|$
(b) Prove that if $\lim _{z \rightarrow a} f(x)=l$ and $\lim _{z \rightarrow a} g(x)=m,$ then $\lim _{x \rightarrow a} \max (f, g)(x)$
$=\max (l, m)$ and similarly for min.

Key Concepts

Maximum and Minimum Functions

The maximum and minimum functions are defined as the operations that select the larger or smaller value respectively from two inputs. These functions are continuous when applied to continuous functions, meaning that the limit of the maximum (or minimum) of two functions is equal to the maximum (or minimum) of their respective limits. This principle allows the evaluation of limits through such composed functions.

Absolute Value Function

The absolute value function is a fundamental example of a continuous function. Its continuity guarantees that if a function converges to a limit l, then the function composed with the absolute value operation will converge to the absolute value of l. This is a specific case of the more general principle that continuous functions can be interchanged with limits.

Limit of a Function

The limit of a function describes the behavior of a function as its input approaches a specific point. This concept is formalized by the epsilon-delta definition, which provides a precise mathematical framework for understanding how function values can be made arbitrarily close to a target limit as the independent variable approaches a given point.

Continuity

Continuity is a property of a function where small changes in the input lead to small changes in the output. In particular, if a function is continuous at a point, the limit of the function as the input approaches that point is equal to the function’s value at that point. This concept plays a key role in applying operations to limits since continuous operations preserve the limit.

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