For the following exercises, suppose that limx →a f(x)=L and limx →a g(x)=M both exist. Use the

step 1 1 -206 if c is equal to 0 then c f of x is equal to 0 so the limit of x tends to a of c f of x i

For the following exercises, suppose that limx →a f(x)=L and...

For the following exercises, suppose that $$\lim _{x \rightarrow a} f(x)=L$$ and $$\lim _{x \rightarrow a} g(x)=M$$ both exist. Use the precise definition of limits to prove the following limit laws: $$\lim _{x \rightarrow a}[c f(x)]=c L$$ for any real constant $c$ (Hint: Consider two cases: $c=0$ and $c \neq 0 .$)

For the following exercises, suppose that $$\lim _{x \rightarrow a} f(x)=L$$ and $$\lim _{x \rightarrow a} g(x)=M$$ both exist. Use the precise definition of limits to prove the following limit laws:
$$\lim _{x \rightarrow a}[c f(x)]=c L$$ for any real constant $c$ (Hint: Consider two cases: $c=0$ and $c \neq 0 .$)

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Key Concepts

Epsilon-Delta Definition of Limits

This is a rigorous formulation of the concept of a limit, stating that a function f(x) approaches a limit L as x approaches a if for every ? > 0 there exists a ? > 0 such that whenever 0 < |x - a| < ?, it follows that |f(x) - L| < ?. This definition underpins all formal proofs in calculus involving limits.

Scaling Law for Limits

The scaling law, or constant multiple law, asserts that if a function f(x) has a limit L as x approaches a, then the function multiplied by a constant, c·f(x), has a limit c·L. This property is proven using the epsilon-delta definition by showing that the behavior of f(x) near a is proportionally scaled by the constant c.

Case Analysis in Limit Proofs

Case analysis is a common technique used in proofs to handle different scenarios that may affect the outcome. In the context of proving the scaling law for limits, separating the proof into the cases where c = 0 and c ? 0 ensures that the argument holds in all instances. For c = 0, the function becomes constant, and for c ? 0, the transformation directly scales the error bound in the epsilon-delta framework.

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