difference rule of limits
The difference rule of limits is a rule used in calculus to find the limit of the difference between two functions
The difference rule of limits is a rule used in calculus to find the limit of the difference between two functions. It states that if the limits of two functions exist as x approaches a certain value, then the limit of their difference also exists and is equal to the difference of their respective limits.
Formally, if lim(x→c) f(x) = L and lim(x→c) g(x) = M, then lim(x→c) [f(x) – g(x)] = L – M.
In other words, if you have two functions f(x) and g(x), and you know their individual limits as x approaches a certain value, you can find the limit of their difference by simply subtracting the limits of the individual functions.
For example, let’s say we have the functions f(x) = 3x^2 + 2x and g(x) = x^2 – 5. We want to find the limit of their difference as x approaches 2.
First, we find the limits of the individual functions:
lim(x→2) f(x) = lim(x→2) (3x^2 + 2x) = 3(2)^2 + 2(2) = 16
lim(x→2) g(x) = lim(x→2) (x^2 – 5) = (2)^2 – 5 = -1
Using the difference rule of limits, we can find the limit of their difference:
lim(x→2) [f(x) – g(x)] = lim(x→2) [3x^2 + 2x – (x^2 – 5)] = lim(x→2) (2x^2 + 2x + 5) = 2(2)^2 + 2(2) + 5 = 17
Therefore, the limit of the difference between f(x) and g(x) as x approaches 2 is 17.
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