Problem 44 - Calculus: Early Transcendentals

Question

Proof of Limit Law 3 Suppose \(\lim _{x \rightarrow a} f(x)=L .\) Prove that \(\lim _{x \rightarrow a}(c f(x))=c L,\) where \(c\) is a constant.

Step-by-Step Solution

Answer

Question: Prove Limit Law 3 which states that if the limit of a function \(f(x)\) approaches \(L\) as \(x\) approaches \(a\), then the limit of a constant times the function \((cf(x))\) approaches \(cL\). Answer: To prove this, we used the formal definition of the limit and showed that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that \(0 < |x-a| < \delta\) implies \(|cf(x) - cL| < \epsilon\). By relating this expression to the given limit definition, we found that \(|cf(x) - cL| = |c||f(x) - L|\) and introduced a new epsilon value, \(\epsilon' = \frac{\epsilon}{|c|}\). This allowed us to show that the limit of the constant times the function approaches the constant times the limit, i.e., \(\lim_{x\rightarrow a}(cf(x)) = cL\).

Step 1: State the formal definition of the limit

Recall the formal definition of the limit: \(\lim_{x\rightarrow a} f(x) = L\) if and only if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that \(0 < |x-a| < \delta\) implies \(|f(x)-L| < \epsilon\).

Step 2: Begin the proof by considering the limit of \(cf(x)\)

We want to prove that \(\lim_{x\rightarrow a}(cf(x)) = cL\). Using the formal definition of the limit, we want to show that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that \(0 < |x-a| < \delta\) implies \(|cf(x)-cL| < \epsilon\).

Step 3: Relate the given limit with the limit we want to prove

Since we know that \(\lim_{x\rightarrow a} f(x) = L\), we know that for a given \(\epsilon > 0\), there exists a \(\delta > 0\) such that \(0 < |x-a| < \delta\) implies \(|f(x)-L| < \epsilon\). We want to relate this expression to the expression we are trying to prove, \(|cf(x)-cL| < \epsilon\). To do that, notice that \(|cf(x)-cL| = |c||f(x)-L|\). Now let \(\epsilon' = \frac{\epsilon}{|c|}\), then we have \(|c||f(x)-L| < \epsilon'\) if \(|c| \neq 0\). If \(|c| = 0\), then \(|c||f(x)-L| = 0\), and the limit law holds trivially.

Step 4: Use the given limit definition with the new epsilon value

We established that \(|c||f(x)-L| < \epsilon'\) for \(|c| \neq 0\). So, given \(\epsilon' > 0\), there exists a \(\delta > 0\) such that \(0 < |x-a| < \delta\) implies \(|f(x)-L| < \epsilon'\). Now, since \(|c||f(x)-L| < \epsilon'\) and \(|c|\epsilon' = \epsilon\), we can see that \(|cf(x)-cL| = |c||f(x)-L| < \epsilon\). Therefore, we have shown that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that \(0 < |x-a| < \delta\) implies \(|cf(x)-cL| < \epsilon\), signifying that \(\lim_{x\rightarrow a}(cf(x)) = cL\).

Key Concepts

Formal Definition of Limit

The formal definition of a limit is the mathematical foundation for understanding how a function behaves near a point. It's a precise way to describe the value that a function's output approaches as the input approaches a certain point.

For a function f(x), we say that the limit of f(x) as x approaches a is L (written as \(\lim_{x\rightarrow a} f(x) = L\)) if, for each quantity (no matter how small), known as epsilon (\(\epsilon > 0\)), there exists a corresponding quantity delta (\(\delta > 0\)) such that if x is within delta distance from a (but not equal to a), then f(x) is within epsilon distance from L. This definition doesn't simply imply closeness; it captures the idea of a function 'tending' towards a value without necessarily reaching it.

Limit of a Function

Understanding the limit of a function means grasping how the function behaves as its input gets closer to some value. For example, as x approaches 0, what does f(x) approach?

The limit tells us about the function's value as x gets infinitely close to a point, but it doesn't have to actually reach that point. Limits are foundational in calculus because they deal with understanding change and instability. They allow us to talk about the slope of a curve (derivative) and the area under a curve (integral), even where those values might not be entirely clear or directly calculable.

Epsilon-Delta Definition

The Epsilon-Delta definition gives a more rigorous and exact meaning to the concept of limits. It quantifies the idea that for each \(\epsilon > 0\), no matter how small, there exists a corresponding \(\delta > 0\) that ensures f(x) is close to L whenever x is close to a.

You can think of \(\epsilon\) as a challenge to show how tightly we can control the output of our function, and \(\delta\) as the solution that proves we can meet that challenge over a particular range of x values near a. This definition is at the heart of proofs in calculus and sets a standard for the precision and rigor expected in mathematical arguments.

Continuity

Continuity is a property of a function that ensures smooth behavior without breaks, jumps, or holes in its graph. A function f(x) is continuous at a point a if the limit of f(x) as x approaches a is equal to f(a). In other words, \(\lim_{x\rightarrow a} f(x) = f(a)\).

This means three things must be true: the function is defined at a (f(a) exists), the limit as x approaches a exists, and these two values are the same. Understanding continuity allows us to analyze functions and guarantees the feasibility of certain operations, like differentiation and integration, which require functions to not have interruptions in their behavior.