Suppose lim⁡x→a f(x)=L{\displaystyle\lim_{x\to a}}\text{ }f\left(x\right)=L and lim⁡x→a g(x)=M{\displaystyle\lim_{x\to a}}\text{ }g\left(x\right)=M. Prove that lim⁡x→a (f(x)−g(x))=L−M{\displaystyle\lim_{x\to a}}\text{ }\left(f\left(x\right)-g\left(x\right)\right)=L-M.

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.7.43

Chapter 2, Problem 2.7.43

Suppose $\lim_{x \to a} f (x) = L$ and $\lim_{x \to a} g (x) = M$ . Prove that $\lim_{x \to a} (f (x) - g (x)) = L - M$ .

Verified step by step guidance

Step 1: Start by recalling the limit properties. One important property is that the limit of a difference is the difference of the limits, provided the individual limits exist. This can be expressed as: if $ \lim_{x \to a} f(x) = L $ and $ \lim_{x \to a} g(x) = M $, then $ \lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) $.

Step 2: Apply the limit property to the given functions $ f(x) $ and $ g(x) $. According to the problem, $ \lim_{x \to a} f(x) = L $ and $ \lim_{x \to a} g(x) = M $. Therefore, using the property, we have $ \lim_{x \to a} (f(x) - g(x)) = L - M $.

Step 3: Verify the conditions for applying the limit property. Ensure that both $ \lim_{x \to a} f(x) $ and $ \lim_{x \to a} g(x) $ exist and are finite, which is given in the problem statement.

Step 4: Conclude that since the conditions are satisfied and the property is applicable, the limit of the difference $ \lim_{x \to a} (f(x) - g(x)) $ is indeed $ L - M $.

Step 5: Summarize the proof by stating that the limit of the difference of two functions is equal to the difference of their limits, as demonstrated by applying the limit property to the given functions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit of a Function

The limit of a function describes the behavior of that function as its input approaches a certain value. Formally, we say that the limit of f(x) as x approaches a is L if, for every small positive number ε, there exists a corresponding small positive number δ such that whenever 0 < |x - a| < δ, it follows that |f(x) - L| < ε. This concept is fundamental in calculus as it lays the groundwork for continuity and differentiability.

Limit Properties

Limit properties are rules that allow us to compute limits of functions more easily. One important property is that the limit of the difference of two functions is the difference of their limits, provided both limits exist. This means that if lim(x→a) f(x) = L and lim(x→a) g(x) = M, then lim(x→a) (f(x) - g(x)) = L - M. Understanding these properties is crucial for proving statements about limits.

Proof Techniques in Calculus

Proof techniques in calculus involve logical reasoning to establish the validity of mathematical statements. Common methods include direct proof, proof by contradiction, and the epsilon-delta definition of limits. In the context of limits, one often uses these techniques to rigorously demonstrate that a limit exists or to derive relationships between limits, such as proving that the limit of a difference equals the difference of the limits.

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