Problem 77 - Calculus of a Single Variable

Question

Proof Prove that $$\lim _{x \rightarrow c} f(x)=L$$ is equivalent to $$\lim _{x \rightarrow c}[f(x)-L]=0$$

Step-by-Step Solution

Answer

The statement \( \lim _{x \rightarrow c} f(x)=L \) is equivalent to \( \lim _{x \rightarrow c}[f(x)-L]=0 \) because as the function f(x) gets closer to the limit L, the expression \(f(x)-L\) gets closer to zero.

Step 1: Understanding of Limit Properties

Recall that the limit as x approaches a value is the value the function approaches. If \( \lim _{x \rightarrow c} f(x)=L \), this means that as x gets closer to c, f(x) gets closer to L.

Step 2: Subtracting Limit L from f(x)

Consider \(f(x) - L\). As x gets infinitely close to c, \(f(x)\) gets infinitely close to L, the difference \(f(x)-L\) would then get infinitely close to 0, which means that \( \lim _{x \rightarrow c}[f(x)-L]=0 \)

Step 3: Summary

So, \( \lim _{x \rightarrow c} f(x)=L \) is proven to be equivalent to \( \lim _{x \rightarrow c}[f(x)-L]=0 \)

Key Concepts

Limit Properties

Limit properties form the cornerstone of understanding when studying calculus. Limits describe how a function behaves as it approaches a particular point.
For instance, let's consider the expression \( \lim_{x \rightarrow c} f(x) = L \). This states that as \( x \) gets very close to \( c \), the function \( f(x) \) approaches a value \( L \).
This concept is fundamental as it defines continuity and helps analyze function behavior even at points not explicitly included in the domain.
Some key properties of limits include:

• Linearity: If \( \lim_{x \rightarrow c} f(x) = L \) and \( \lim_{x \rightarrow c} g(x) = M \), then \( \lim_{x \rightarrow c} [af(x) + bg(x)] = aL + bM \) for any constants \( a \) and \( b \).
• Product Rule: \( \lim_{x \rightarrow c} [f(x) \cdot g(x)] = L \cdot M \).
• Quotient Rule: If \( M eq 0 \), \( \lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{L}{M} \).

These properties are valuable tools that facilitate manipulating and solving limit problems.

Equivalence of Limits

When we talk about the equivalence of limits, we refer to the concept that if two limit expressions evaluate to the same value, they reflect the same limit behavior. This is precisely what we need to demonstrate with:
- \( \lim _{x \rightarrow c} f(x) = L \) - and \( \lim _{x \rightarrow c} [f(x) - L] = 0 \).The logic is straightforward. If \( f(x) \) approaches \( L \) as \( x \) approaches \( c \), then logically, the function \( f(x) - L \) should approach 0 when \( x \) approaches \( c \) as well. This means:

• Both expressions consider the behavior of \( f(x) \) near \( c \).
• The difference \( f(x) - L \) getting close to 0 confirms that \( f(x) \) itself gets close to \( L \).

Thus, the equivalence of these two limits is intrinsic and backed by the continuity of the limit process.

Limit Proof

To prove the equivalence of the limits \( \lim _{x \rightarrow c} f(x) = L \) and \( \lim _{x \rightarrow c} [f(x) - L] = 0 \), we can follow a simple step-by-step logical argument.
Start by considering \( f(x) \) reaching \( L \) as \( x \) nears \( c \).
Thus, for any chosen closeness \( \epsilon > 0 \), there exists a margin \( \delta > 0 \) such that whenever \( 0 < |x - c| < \delta \), we have \( |f(x) - L| < \epsilon \).
Now, think about \( f(x) - L \). If \( |f(x) - L| < \epsilon \), it implies \( |f(x) - L - 0| < \epsilon \).
This shows that as \( x \) approaches \( c \), \( f(x) - L \) indeed approaches 0.
In this proof, every step solidifies the idea that limiting towards \( 0 \) for \( f(x) - L \) is equivalent to \( f(x) \) reaching \( L \).
Hence, the original and altered expressions are confirmed to convey the same mathematical reality.