You've studied the definition of limits and how to find them algebraically and graphically. Is there perhaps a faster way to find the limit of a function. Yes, there is! We can use limit laws. Here you will see some of the more common properties of limits of functions and how to apply them.
Importance of Limit Laws
You may ask yourself why limit laws in calculus are important. You already know the definition of the limit of a function. Why not just apply that? The reason is that it is much more efficient to prove one thing about functions in general than to use the definition on each and every function. It is the difference between proving that dogs like to play and proving that my dog likes to play, your dog likes to play, the neighbor's dog likes to play... and on and on and on.
Limit Laws in Calculus
Many textbooks will mention the properties of limits listed below since they are the most common ones. Sometimes they will even refer to them as the 5 limit laws.
Theorem: Properties of Limits, also known as Limit Laws
Suppose that \(L\), \(M\), \(a\) and \(k\) are real numbers, with \(f\) and \(g\) being functions such that:
\[lim_{x \rightarrow a} f(x)=L\]
and
\[lim_{x \rightarrow a} g(x)=M\]
Then the following hold:
Sum Rule: \(lim_{x \rightarrow a} (f(x)+g(x))=L+M\)
Difference Rule: \(lim_{x \rightarrow a} (f(x)-g(x))=L-M\)
Product Rule: \(lim_{x \rightarrow a} (f(x)+ \cdot g(x))=L \cdot M\)
Constant Multiple Rule: \(lim_{x \rightarrow a} k \cdot f(x) = k \cdot L\)
Quotient Rule: If \neq 0\) then
\[lim_{x \rightarrow a} \dfrac{f(x)}{g(x)}=\dfrac{L}{M}\]
Power Rule: If \(r, s \in R \), with \(s \neq 0\), then
\[lim_{x \rightarrow a} \left( f(x) \right)^{r/s}=L^{r/s} \]
provided that \(L^{r/s}\) is a real number and \(L>0\) when \(s\) is even.
For more examples of how to find limits of particular functions, see Finding Limits. For a reminder on the definition of the limit of a function, see Limits of a Function.
It is essential to make sure that the conditions are met before applying the properties of limits. Let's see an example.
Take
\(f(x)=x\) and \(g(x)=\dfrac{1}{x}\) and try to find:
\[lim_{x \rightarrow 0} (f \cdot g)(x)\].
Answer:
You are probably tempted just to use the Product Rule for limits. You already know that
\[lim_{x \rightarrow 0} f(x)=0\]
However, if you try and apply the definition of the limit for \(g(x)\), you can see that no matter how close you take your \(\delta\) window to be to \(a=0\), it won't work because the function has a vertical asymptote at \(x=0\). So, \(g(x)\) doesn't have a limit at \(a=0\). But
\[lim_{x \rightarrow 0} (f(x) \cdot g(x))= lim_{x \rightarrow 0} \left( x \cdot \dfrac{1}{x} \right)\]
\[lim_{x \rightarrow 0} (f(x) \cdot g(x))=lim_{x \rightarrow 0} (1)\]
\[lim_{x \rightarrow 0} (f(x) \cdot g(x))=1\]
which is not what you get when you multiply together \(0\) and something that doesn't exist! So while the limit of the product does exist, the product of the limits does not.
Calculating Limits Using the Limit Laws
For some functions, the limit laws get used so much that it is easier to look at kinds of functions rather than at lots of functions. It turns out that polynomials and rational functions are especially nice.
Definitions and Limit Laws
In the following examples, the definition of the limit was used to show that
\[lim_{x \rightarrow a} x=a\]
and
\[lim_{x \rightarrow a} k=k\]
where \(k\) is a constant. See Limits of a Function for more details on how to apply the definition of the limit.
Take the function
\[f(x)=10x^3-2x+1\]
and \(a\) to be a constant real number. Find
\[lim_{x \rightarrow a} f(x)\]
Answer:
Looking carefully, you can notice that the function is just the sum and product of powers of \(x\), along with the constant! So, the condition required to use our limit laws is met! Applying them gives:
\[lim_{x \rightarrow a} f(x) = lim_{x \rightarrow a} (10x^3-2x+1)\]
\[lim_{x \rightarrow a} f(x)= 10 (lim_{x \rightarrow a} x) (lim_{x \rightarrow a} x) (lim_{x \rightarrow a} x)- 2 (lim_{x \rightarrow a} x)+ (lim_{x \rightarrow a} 1) \]
\[10a^3-2a+1\]
In the example above, you looked at a specific polynomial and found the limit exists. It turns out that you can do this same process (using the Sum Rule, Constant Rule, and the Power Rule) to find the limit of any polynomial!
If \(f(x)\) is a polynomial and \(a\) is a real number, then
\[lim_{x \rightarrow a} f(x)=f(a)\]
Limits of Rational Functions
Taking the limit of rational functions can sometimes be a challenge. For more examples of techniques you can use for rational functions, see Finding Limits of Specific Functions.
In the case where the point you want to take the limit at is in the domain of the rational function, taking the limit is not difficult. Take
\[f(x)= \dfrac{x^2+3}{x-1}\]
and find the limit as \(x \rightarrow 4\).
Answer:
First, ask yourself if \(x=4\) is in the domain of the function. It turns out that it is, and in fact
\[f(4)=\dfrac{4^2+3}{4-1}=\dfrac{16+3}{3}=\dfrac{19}{3}\].
So using the Quotient Rule tells you that
\[lim_{x \rightarrow 4} = \dfrac{19}{3}\]
Similar to the result for polynomials, you can say the following about rational functions:
If \(f(x)\) is a rational function and \(a\) is a real number in the domain of \(f(x)\), then
\[lim_{x \rightarrow a} f(x)=f(a)\]
Examples Using the Limit Laws
Rather than looking at a function with a definition, sometimes all you will know are some properties of the functions involved, and you will need to use Limit Laws to draw conclusions about the functions.
Suppose that
\(lim_{x \rightarrow 7} f(x)=3\) and \(lim_{x \rightarrow 7} g(x) =-1\)
If possible, find the following:
\[lim_{x \rightarrow 7} (f+g)(x)\]
\[lim_{x \rightarrow 7} 4g(x)\]
\[lim_{x \rightarrow 7} \left( \dfrac{f}{g} \right)(x)\]
and
\[lim_{x \rightarrow 7} \sqrt{g(x)}\]
Answer:
1. To find
\[lim_{x \rightarrow 7} (f+g)(x)\]
you have all the conditions satisfied to apply the Sum Rule, so
\[lim_{x \rightarrow 7} (f+g)(x)=lim_{x \rightarrow 7} f(x) + lim_{x \rightarrow 7} g(x)\]
\[lim_{x \rightarrow 7} (f+g)(x)=3+(-1)\]
\[lim_{x \rightarrow 7} (f+g)(x)=2\]
2. You can use the Constant Rule for the next one, so
\[lim_{x \rightarrow 7} 4g(x)=4 lim_{x \rightarrow 7} g(x)\]
\[lim_{x \rightarrow 7} 4g(x)=4(-1)\]
\[lim_{x \rightarrow 7} 4g(x)=-4\]
3. Since the limit of \(g(x)\) as \(x \rightarrow 7\) is not equal to zero, you can apply the Quotient Rule to see that
\[lim_{x \rightarrow 7} \left( \dfrac{f}{g} \right)=\dfrac{\lim_{x \rightarrow 7}f(x)}{lim_{x \rightarrow 7} g(x)}=\dfrac{3}{-1}=-3\]
4. The last one is a bit more challenging. Here,
\[lim_{x \rightarrow 7} \sqrt{g(x)}=lim_{x \rightarrow 7}\sqrt{-1}\]
You cannot take the square root of a negative number and get a real number back, so you can't evaluate this. That means you can't find
\[lim_{x \rightarrow 7} \sqrt{g(x)}\]
Limit Laws - Key takeaways
Suppose that \(L\), \(M\), \(a\) and \(k\) are real numbers, with \(f\) and \(g\) being functions such that:
\(lim_{x \rightarrow a} f(x)=L\) and \(lim_{x \rightarrow a} g(x)=M\)
Then the following hold:
Sum Rule: \(lim_{x \rightarrow a} (f(x)+g(x))=L+M\)
Difference Rule: \(lim_{x \rightarrow a} (f(x)-g(x))=L-M\)
Product Rule: \(lim_{x \rightarrow a} (f(x) \cdot g(x))L \cdot M\)
Constant Multiple Rule: \(lim_{x \rightarrow a} k \cdot f(x)=k \cdot L\)
Quotient Rule: If \(M \neq 0\) then \[lim_{x \rightarrow a} \dfrac{f(x)}{g(x)}=\dfrac{L}{M}\]
Power Rule: If \(r, s, \in Z\), with \(s \neq 0\), then \[lim_{x \rightarrow a} (f(x))^{r/s}=L{r/s}\]
provided that \(L^{r/s}\) is a real number and \(L>0\) when \(s\) is even.
- If \(f(x)\) is a polynomial and \(a\)is a real number, then \[lim_{x \rightarrow a} f(x)=f(a)\]
If \(f(x)\) is a rational function and \(a\) is a real number in the domain of \(f(x)\), then \[lim_{x \rightarrow a} f(x)=f(a)\]
Always be sure that the conditions to use one of the Limit Laws are met before you use it!
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