Mastering the Sum Rule of Limits: Calculus Made Easy

Sum Rule of Limits

The Sum Rule of Limits is a property used in calculus to evaluate the limit of the sum of two functions

The Sum Rule of Limits is a property used in calculus to evaluate the limit of the sum of two functions. It states that if the limits of two functions exist, then the limit of their sum is equal to the sum of their limits.

Formally, let f(x) and g(x) be two functions, and let L and M be real numbers such that:

lim(x->a) f(x) = L and lim(x->a) g(x) = M

Then, the sum of the limits can be written as:

lim(x->a) [f(x) + g(x)] = L + M

This means that if both limits L and M exist, then the limit of their sum is simply the sum of the limits.

Here’s an example to illustrate the Sum Rule of Limits:

Let’s consider the functions f(x) = 2x and g(x) = 3x^2. We want to find the limit of their sum as x approaches 2.

First, let’s find the limits of the individual functions:
lim(x->2) 2x = 2 * 2 = 4
lim(x->2) 3x^2 = 3 * 2^2 = 12

Now, we can use the Sum Rule of Limits to find the limit of their sum:
lim(x->2) [f(x) + g(x)] = lim(x->2) [2x + 3x^2]

Using the Sum Rule, we can simply add the limits:
lim(x->2) [2x + 3x^2] = 4 + 12 = 16

Therefore, the limit of the sum of the functions f(x) = 2x and g(x) = 3x^2 as x approaches 2 is equal to 16.

The Sum Rule of Limits makes evaluating limits of sums of functions much simpler, as we only need to find the limits of the individual functions and then add them together.

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