The Sum Rule of Limits: Finding the Limit of a Sum of Functions in Calculus

Sum Rule of Limits

The sum rule of limits is a fundamental principle in calculus that allows us to find the limit of a sum of functions

The sum rule of limits is a fundamental principle in calculus that allows us to find the limit of a sum of functions. Specifically, it states that if two functions have limits as x approaches a point, then the limit of their sum is the sum of their limits.

Mathematically, suppose we have two functions f(x) and g(x), and we know that as x approaches a particular value (let’s call it c), both f(x) and g(x) approach certain limits. Let’s denote these limits as L and M respectively.

Then, according to the sum rule of limits, we can say that:

lim(x→c) [f(x) + g(x)] = lim(x→c) f(x) + lim(x→c) g(x) = L + M

In other words, to find the limit of the sum of two functions, we can simply calculate the limits of each function separately and then add them together.

It is important to note that this rule holds true whether the limits L and M are finite numbers, positive or negative infinity, or even if they do not exist. However, the limits of f(x) and g(x) must exist individually for the sum rule of limits to be applicable.

Let’s illustrate this with an example:

Example 1: Find the limit of f(x) + g(x) as x approaches 3, given that:

lim(x→3) f(x) = 4
lim(x→3) g(x) = -2

Using the sum rule of limits, we can add these two limits together:

lim(x→3) [f(x) + g(x)] = lim(x→3) f(x) + lim(x→3) g(x) = 4 + (-2) = 2

Hence, the limit of f(x) + g(x) as x approaches 3 is 2.

It is worth mentioning that the sum rule of limits can be extended to more than two functions. If we have three functions f(x), g(x), and h(x), and we know the limits of each as x approaches a certain value, then we can simply add these limits together:

lim(x→c) [f(x) + g(x) + h(x)] = lim(x→c) f(x) + lim(x→c) g(x) + lim(x→c) h(x)

And so on, for any finite number of functions.

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