Question

Prove that \(\lim f(x)=L\) is equivalent to \(\lim [f(x)-L]=0\).

Short answer

The statement \(\lim_{x \to a} f(x) = L\) is indeed equivalent to \(\lim_{x \to a} [f(x) - L] = 0\). This has been proven in two directions, first showing that if the limit is \(L\) then the limit of the function minus \(L\) is \(0\), and then showing that if the limit of the function minus \(L\) is \(0\), then the limit of the function is \(L\).

Step-by-step solution

Assumption

Assume that the limit of a function \(f(x)\) as \(x\) approaches a certain value, is \(L\). Mathematically, this can be expressed as, \(\lim_{x \to a} f(x) = L\).

Subtract \(L\)

From what we know about the properties of limits, the limit of a function \(f(x)\) minus a constant as \(x\) approaches a certain value, equals to the limit of \(f(x)\) as \(x\) approaches that value, minus the constant. Thus, if we subtract \(L\) from both sides of our known limit \(\lim_{x \to a} f(x) = L\), we get that \(\lim_{x \to a} [f(x) - L] = L - L\), which simplifies to \(\lim_{x \to a} [f(x) - L] = 0\).

Proof the other direction

For the other direction, assume that \(\lim_{x \to a} [f(x) - L] = 0\). Knowing that limits of a function minus a constant can be taken by subtracting the constant from the limit of the function, we can rewrite the expression to \(\lim_{x \to a} f(x) - L = 0\). Adding \(L\) to both sides, gives \(\lim_{x \to a} f(x) = L\) thus completing the proof.

Key concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It's divided into two main parts: differential calculus and integral calculus. Differential calculus concerns the concept of instantaneous rate of change, while integral calculus is about accumulation of quantities.

In the context of the textbook exercise, calculus plays a crucial role in understanding the behavior of functions as variables approach certain points. Limits, which are fundamental to calculus, describe the value that a function 'approaches' as the input (or 'x' value) approaches some value.

When we talk about the limit of a function, we are referring to the value that the function approaches as 'x' approaches a specific number. The concept of a limit is essential because it allows mathematicians and scientists to determine the exact behavior of functions at points where they may not be expressly defined, such as points of discontinuity or where there is an asymptotic behavior.

Limits Properties

When dealing with the limits of functions in calculus, there are several properties that are inherently useful in simplifying complex expressions and proving certain mathematical statements. One of these properties is the limit of a constant function: if 'c' is a constant, then \( \lim_{x \to a} c = c \).

Similarly, the limit of a function plus or minus a constant can be deduced by adding or subtracting the constant after the limit has been evaluated by breaking down complex expressions into simpler components.

Algebra of Limits

The properties of limits follow the algebra of limits which states that the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and the limit of a quotient is the quotient of the limits (provided the limit of the denominator is not zero).

In the exercise provided, this property of limits is exploited to show that \( \lim f(x)=L \) is indeed the same as \( \lim [f(x)-L]=0 \). By understanding and applying these properties, computations involving limits become systematic and problems can be solved with a set protocol of steps.

Proof Techniques

Proof techniques are systematic approaches employed to establish the truth of mathematical statements. Amongst various approaches, direct proof, contradiction, and contraposition are common.

For limits, particularly, direct proof—which involves assuming certain definitions and logically deducing the desired result—is commonly used. This method depends on applying known properties and theorems to reach a conclusion that is logically consistent with the premises.

Using Direct Proof in Limits

In the textbook solution, direct proof is used to show the equivalence of the statements \( \lim f(x)=L \) and \( \lim [f(x)-L]=0 \). We start with the assumption that the initial statement is true, apply known properties about limits to manipulate the function and establish the equivalence.

Additionally, this technique often involves comparative steps that address the statement both forwards and backwards. This two-way approach solidifies the overall understanding of the limit in question and verifies the result from both perspectives, showing mathematical equivalence and reinforcing the connection between the premise and the conclusion.