Properties of Limits

In order to evaluate limits, we can take a look at their various properties to help simplify the process. In this instance, assume that both \(f\) and \(g\) have limits that exist at \(x = a\) and that \(c\) represents the constant.

The properties can be summarized as such:

Constant \(\lim\limits_{x\to a}c=c\)
Constant times a Function \(\lim\limits_{x\to a}[c \cdot f(x)]= c\lim\limits_{x\to a}f(x)\)
Sum of Functions \(\lim\limits_{x\to a}[f(x)+g(x)] = \lim\limits_{x\to a}f(x)+\lim\limits_{x\to a}g(x)\)
Difference of Functions \(\lim\limits_{x\to a}[f(x)-g(x)] = \lim\limits_{x\to a}f(x)-\lim\limits_{x\to a}g(x)\)
Product of Functions \(\lim\limits_{x\to a}[f(x)\cdot g(x)] = \lim\limits_{x\to a}f(x)\cdot\lim\limits_{x\to a}g(x)\)
Quotient of Functions \(\lim\limits_{x\to a} \left[\cfrac{f(x)}{g(x)}\right] = \cfrac{\lim\limits_{x\to a}f(x)}{\lim\limits_{x\to a}g(x)}\)
Function Raised to an Exponent \(\lim\limits_{x\to a}[f(x)]^n=\lim\limits_{x\to a}[f(x)]^n\)
nth root of a Function, 0 < n \(\lim\limits_{x\to a}\sqrt[n]{f(x)} = \sqrt[n]{\lim\limits_{x\to a}f(x)}\)
Polynomial Function \(\lim\limits_{x\to a}p(x)=p(a)\)


Here are some additional pointers for helping determine the limits for each expression:

  • Substitute what \(x\) approaches right away. If defined, that will be the answer
  • If you have the limit in the indeterminate form \(\cfrac{0}{0}\), you can factor, expand, or find the Lowest Common Denominator. You can also rationalize the numerator and denominator by the same value to make the denominator a whole number.
  • If you have the limit in the form \(\cfrac{\text{Real Number}}{0}\) or have it in a single square root, then try numbers close to the point on both sides to show the limit won't exist
  • If you have a piecewise or absolute value function, you must do both sides of the limit separately
  • If you have \(\lim\limits_{x\to∞}\) with rational numbers in both the numerator and denominator, divide by the highest power

Example

Evaluate the following limits:

  1. \(\lim\limits_{x\to5}\sqrt{\cfrac{x^2}{x-1}}\)

  2. \(\lim\limits_{x\to3}\sqrt{9-x^2}\)

i. First, as the entire function is placed under a square root, we can use the Function Raised to Exponent Property to determine that its being raised to a power of \( \frac{1}{2} \).

Next, we can use the Quotient Property to find the limits of the respective pieces:

\(= \sqrt{\cfrac{\lim\limits_{x\to5}x^2}{\lim\limits_{x\to5}x-1}}\)

Then, we can use the Function Raised to an Exponent Property in the numerator to substitute the limit for \(x\):

\(= \sqrt{\cfrac{(\lim\limits_{x\to5}x)^2}{\lim\limits_{x\to5}x-1}}\)

Finally, we can use the Constant Property to substitute the limit value (in this case, \(5\)) for \(x\). After, we can simplify to get the final result:

\(= \sqrt{\cfrac{(5)^2}{5-1}}\)

\(= \sqrt{\cfrac{25}{4}}\)

\(= \cfrac{5}{2}\)

Therefore, we can determine that the limit exists.


ii. First, we can test the limit from its left side:

\(= \lim\limits_{x\to3^{-}}\sqrt{9-x^2}\)

We can subsitute a value close to the limit (in this case \(2.9\) for \(x\). We can then simplify the function to evaluate the limit:

\(= \sqrt{9-(2.9)^2}\)

\(= \sqrt{9-8.41}\)

\(= \sqrt{0.59}\)

\(= 0.77\)

We can determine that testing the limit from its left side provides us with a real value.

Next, we can evaluate the limit from its right side:

\(= \lim\limits_{x\to3^{+}}\sqrt{9-x^2}\)

We can substitute this value for \(x\) in the function. We can then simplify it to evaluate the limit:

\(= \sqrt{9-(3.1)^2}\)

\(= \sqrt{9-9.61}\)

\(= \sqrt{-0.61}\)

\(= \text{UNDEFINED}\)

When we compare the limits on both sides, we get the following result:

\(\lim\limits_{x\to3^{-}} \neq \lim\limits_{x\to3^{+}}\)

Therefore, we can determine that this limit does not exist as it only represents a one-sided limit. Since the domain is restricted, we can determine that the function doesn't exist on the left side.


Evaluate the following limits.

\(\lim\limits_{x\to2}\cfrac{|2x-4|}{x-2}\)


\(\lim\limits_{x\to∞}\cfrac{4 - 3x^3 + 2x}{8 - x^2 + 7x^3}\)


\(\lim\limits_{x\to0}\cfrac{\sqrt{4+x}-2}{x}\)


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