The Difference Rule of Limits: Finding the Limit of the Difference between Two Functions in Calculus

difference rule of limits

The difference rule of limits, also known as the subtraction rule, is a rule in calculus that allows us to find the limit of the difference between two functions

The difference rule of limits, also known as the subtraction rule, is a rule in calculus that allows us to find the limit of the difference between two functions. It states that if the limits of two functions exist as x approaches a, then the limit of their difference also exists and is equal to the difference of their limits.

Mathematically, if lim(x→a) f(x) = L and lim(x→a) g(x) = M, then lim(x→a) [f(x) – g(x)] = L – M.

To prove this rule, we can use the definition of limits. Let’s assume that lim(x→a) f(x) = L and lim(x→a) g(x) = M. We need to show that lim(x→a) [f(x) – g(x)] = L – M.

By the definition of limits, we know that for any epsilon greater than 0, there exists a delta1 greater than 0 such that if 0 < |x - a| < delta1, then |f(x) - L| < epsilon. Similarly, since lim(x→a) g(x) = M, for any epsilon greater than 0, there exists a delta2 greater than 0 such that if 0 < |x - a| < delta2, then |g(x) - M| < epsilon. Now we want to find a delta such that if 0 < |x - a| < delta, then |[f(x) - g(x)] - (L - M)| < epsilon. Since |[f(x) - g(x)] - (L - M)| = |f(x) - L + M - g(x)|, we can use the triangle inequality to get: |[f(x) - g(x)] - (L - M)| ≤ |f(x) - L| + |M - g(x)|. Now, we can choose delta to be the minimum of delta1 and delta2. Then, for 0 < |x - a| < delta, we have: |f(x) - L| < epsilon, |M - g(x)| < epsilon. Therefore, |[f(x) - g(x)] - (L - M)| ≤ |f(x) - L| + |M - g(x)| < 2epsilon. Hence, we have shown 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. This difference rule of limits can be very handy when dealing with limits in calculus, as it allows us to simplify expressions and evaluate limits more easily.

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