
It’s almost time for us to calculate some limits. However, before doing so, we need some properties of limits that make our life somewhat easier. So, let’s look at them first.
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Step by step guide to properties of limits
The following is the list of properties of limits:
We assume that \( lim_{x\to a}f(x)\) and \( lim_{x\to a}g(x)\) exist and \(c\) is a constant. Then,
- \(\color{blue}{lim_{x\to a}[c . f(x)]=c\ lim_{x\to a} f(x)}\)
You can factor a constant that is multiplicative out of a limit.
- \(\color{blue}{lim_{x\to a}[f(x) ± g(x)]=lim_{x\to a}f(x) ± lim_{x\to a}g(x)}\)
To consider the limit of a sum of difference, select the limits individually and put them back with the corresponding sign. This fact works regardless of the number of functions we separated by \(+\) or \(-\).
- \(\color{blue}{lim_{x\to a}[f(x) . g(x)]=lim_{x\to a} f(x) . lim_{x\to a} g(x)}\)
Consider product limits similar to sums or differences. just select the limit of the pieces and put them back, and this is not limited to just two functions.
- \(\color{blue}{lim_{x\to a}[\frac{f(x)}{g(x)}]=\frac{lim_{x\to a} f(x)}{lim_{x\to a} g(x)}}\), \(\color{blue}{lim_{x\to a} g(x) ≠0}\)
we need to bother only if the limit of the denominator is zero when operating the quotient limit. If it were zero, it ends up with a division by zero error.
- \(\color{blue}{lim_{x\to a}[f(x)]^n]=[lim_{x\to a}f(x)]^n}\), where \(n\) is any real number
In this property \(n\) can be any real number (positive, negative, integer, fraction, irrational, zero).
- \(\color {blue}{lim_{x\to a}[\sqrt[n]{f\left(x\right)}]=\sqrt[n]{lim_{x\to a}f(x)}}\)
- \(\color{blue}{lim_{x\to a} c= c}\)
The limit of a constant is only a constant.
- \(\color{blue}{lim_{x\to a}x=a}\)
- \(\color{blue}{lim_{x\to a}x^n=a^n}\)
Note: all these properties also hold for the two one-sided limits as well we just didn’t write them down with one-sided limits to save on space.
Properties of Limits – Example 1:
Calculate the value of the following limit. \(lim_{x\to -2}(4x^2+3x-2)\)
First, we use this property to break up the limit into three separate limits: \(\color{blue}{lim_{x\to a}[f(x) ± g(x)]=lim_{x\to a}f(x) ± lim_{x\to a}g(x)}\)
\(lim_{x\to -2}(4x^2+3x-2)\) \(=lim_{x\to -2}4x^2+ lim_{x\to -2}3x- lim_{x\to -2}2\)
Then, use this property to bring the constants out of the first two limits: \(\color{blue}{lim_{x\to a}[c . f(x)]=c\ lim_{x\to a} f(x)}\)
\(=lim_{x\to -2}4x^2+ lim_{x\to -2}3x- lim_{x\to -2}2\) \(=4lim_{x\to -2}x^2+ 3lim_{x\to -2}x- lim_{x\to -2}-2\)
Now, use these property to solve limit:
\(\color{blue}{lim_{x\to a} c= c}\) and \(\color{blue}{lim_{x\to a}x^n=a^n}\)
\(=4lim_{x\to -2}x^2+ 3lim_{x\to -2}x- lim_{x\to -2}-2\)
\(=4(-2)^2+3(-2)-2\)
\(=16-6-2=8\)
Properties of Limits – Example 2:
Calculate the value of the following limit. \(lim_{x\to 3}(4x^2)\)
To find the limit, use this formula: \(\color{blue}{lim_{x\to a}[c . f(x)]=c\ lim_{x\to a} f(x)}\)
\(lim_{x\to 3}(4x^2)\) \(=4 lim_{x\to 3} x^2\)
\(= 4(3)^2=4(9)\)
\(=36\)
Exercises for the Properties of Limits
Calculate the value of the following limit.
- \(\color {blue}{lim_{x\to 2}(2x+3)^4}\)
- \(\color{blue}{lim_{x\to 3}\frac{x^2-x-6}{x-3}}\)
- \(\color{blue}{lim _{x\to 4}\left(\frac{\frac{1}{x}-\frac{1}{4}}{x-4}\right)}\)
- \(\color{blue}{lim _{x\to 7}\left(\frac{x^2+7x+12}{x+7}\right)}\)
- \(\color{blue}{lim _{x\to 0}\left(\frac{\sqrt{36-x}-6}{x}\right)}\)

- \(\color{blue}{2401}\)
- \(\color{blue}{5}\)
- \(\color{blue}{-\frac{1}{16}}\)
- \(\color{blue}{\frac{55}{7}}\)
- \(\color{blue}{-\frac{1}{12}}\)
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