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This article was written in collaboration with wikiHow writer Devin McSween. Devin McSween is a wikiHow staff writer. With a background in psychology, he has presented his research in social psychology at various conferences and has contributed several manuscripts to his publication. At wikiHow, Devin combines his love of writing and research with the goal of providing wikiHow readers with accessible information to help them learn and grow. He graduated from the University of Charleston with a degree in psychology. There are 8 references cited in this article, which can be found at the bottom of the page. This article has been viewed 1212 times. Know more…
Just when you start to understand limits, your teacher tells you that sometimes they don’t exist. There should be an easy way to determine that the limit does not exist, but how? Well, we’ve got you covered! In this article, we will look at 4 clear cases where there is no restriction and tell you how to find where there are no restrictions for different features. If you’re ready to dig into the limitations, keep reading!
- The limit does not exist when the left and right sides of the function approach different values.
- If the function approaches negative or positive infinity as it approaches a value, or if it oscillates between multiple values, the limit does not exist.
- Find where the limit does not exist by plotting the function by hand or on a calculator.
The limits are different on each side of the function. When evaluating the limit of a function, notice how x{\displaystyle x} approaches the value c{\displaystyle c} on the left and right side of the function. If the left side of the function approaches a different limit than the right side, then the limit does not exist. This means that the function is not continuous at all times, which usually happens when there is a jump or gap in the graph of the function.[1]
- For example, look at the graph limx→0|x|x{\displaystyle \lim _{x\to 0}{\frac {|x|}{x}}}.
- as x{\displaystyle x} approaches 0 from the left, approaches y=−1{\displaystyle y=-1}
- as x{\displaystyle x} approaches 0 from the right, approaches y=1{\displaystyle y=1}
- The limits of the left and right sides cannot be distinguished for the limit to exist, so limx→0|x|x{\displaystyle \lim _{x\to 0}{\frac {|x|}{x }}} there is no.
- The left limit is written as limx→c−f(x)=L{\displaystyle \lim _{x\to c-}f(x)=L}where the limit of f(x){\displaystyle f(x)} as x{\displaystyle x} approaches the value c{\displaystyle c} is the limit of L{\displaystyle L}. On the left you are looking at the values of x{\displaystyle x} which are less than c{\displaystyle c}.
- The right limit is written as limx→c+f(x)=L{\displaystyle \lim _{x\to c+}f(x)=L}. On the right hand edge, you are looking at the x{\displaystyle x} values which are greater than c{\displaystyle c}.
- For example, look at the graph limx→0|x|x{\displaystyle \lim _{x\to 0}{\frac {|x|}{x}}}.
The function is infinite or does not approach a finite value. Some functions have curves that approximate a vertical line, called a vertical asymptote. The function never touches the line, but the distance between the curve and the line gets closer and closer to 0 as the function moves toward positive or negative infinity.[2]
If you evaluate limx→c{\displaystyle \lim _{x\to c}} and the function has a vertical asymptote at x=c{\displaystyle x=c}, then the limit does not exist. This is because at least one side of the function approaches infinity at x=c{\displaystyle x=c}which is not a finite real number.[3]- For example, look at the graph limx→01×2{\displaystyle \lim _{x\to 0}{\frac {1}{x^{2}}}}.
- This function has a vertical asymptote at x=0{\displaystyle x=0}.
- The left hand side of the function approaches infinity as it approaches x=0{\displaystyle x=0}then limx→0−1×2=∞{\displaystyle \lim _{x\to 0-}{\frac {1}{x^{2}}}=\infty }.
- The right hand side of the function approaches infinity as it approaches x=0{\displaystyle x=0}then limx→0+1×2=∞{\displaystyle \lim _{x\to 0+}{\frac {1}{x^{2}}}=\infty }.
- So, limx→01×2=∞{\displaystyle \lim _{x\to 0}{\frac {1}{x^{2}}}=\infty } which is not a finite number, so it has no limit.
- For example, look at the graph limx→01×2{\displaystyle \lim _{x\to 0}{\frac {1}{x^{2}}}}.
The function oscillates between more than 1 value. For the limit to hold, the function must stop at a value of 1 as it approaches some value c{\displaystyle c}. Sometimes the function tends to bounce or oscillate between 2 or more values as it approaches c{\displaystyle c}. As c {\displaystyle c} approaches, the oscillation becomes faster. In these cases the function is not set to 1 value, so there is no limit.[4]
- For example, look at the graph limx→0cos(πx){\displaystyle \lim _{x\to 0}cos({\frac {\pi }{x}})}.
- On both sides of the function, such as x{\displaystyle x} approaches 0, ranging widely between y=−1{\displaystyle y=-1} iy=1{\displaystyle y=1}.
- Jumping back and forth around 0 means that the limit does not exist for limx→0cos(πx){\displaystyle \lim _{x\to 0}cos({\frac {\pi }{x}})}.
- For example, look at the graph limx→0cos(πx){\displaystyle \lim _{x\to 0}cos({\frac {\pi }{x}})}.
The function is defined only for some values of x. Remember how the border of the left and right sides must approach the same value for the border to exist? Some functions have values of x that are undefined or do not exist. If the function cannot access some value c{\displaystyle c} on side 1, since the values of x do not exist, then the limit for that function cannot exist.[5]
- For example, look at the graph limx→0x{\displaystyle \lim _{x\to 0}{\sqrt {x}}}.
f(x)=x{\displaystyle f(x)={\sqrt {x}}} is undefined for any value of x less than 0 because you can’t take the square root of a negative number (it gives you an imaginary number).
- So as long as x{\displaystyle x} approaches 0 from the right, x{\displaystyle x} cannot access any value to the left because there is no value of x below 0. Therefore, limx→0x{\displaystyle \lim _{x\to 0}{\sqrt {x}}} there is no.
- For example, look at the graph limx→0x{\displaystyle \lim _{x\to 0}{\sqrt {x}}}.
approaches the value c{\displaystyle c}
on the left and right side of the function. If the left side of the function approaches a different limit than the right side, then the limit does not exist. This means that the function is not continuous at all times, which usually happens when there is a jump or gap in the graph of the function.[1]
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where the limit of f(x){\displaystyle f(x)}
as x{\displaystyle x}
. On the left you are looking at the values of x{\displaystyle x}
. On the right hand edge, you are looking at the x{\displaystyle x} values
and the function has a vertical asymptote at x=c{\displaystyle x=c}
, then the limit does not exist. This is because at least one side of the function approaches infinity at x=c{\displaystyle x=c}
.
.
.
.
which is not a finite number, so it has no limit.
.
.
is undefined for any value of x less than 0 because you can’t take the square root of a negative number (it gives you an imaginary number).Graph the function and see how the left and right sides converge. C.{\ display style c}. The easiest way to estimate the limit is to look at the behavior of the graph as x{\displaystyle x} approaches some value c{\displaystyle c}. Graph the function by hand or use a scientific calculator to graph it. Then look at the left and right focus. Do they approach different values? Does one side go to infinity? Does the function oscillate between several values? If so, there is no limit.[6]
- For example, calculate limx→52x−5{\displaystyle \lim _{x\to 5}{\frac {2}{x-5}}}.
- Draw a graph on paper or enter the function on your calculator. On most scientific calculators, press the button “AND =” and enter your function. Then press the button “graphic”.
- Notice how the left and right sides of the function approximate c=5{\displaystyle c=5}. The left side goes toward negative infinity as it gets closer, while the right side goes toward positive infinity. So there is no limit.
- For example, calculate limx→52x−5{\displaystyle \lim _{x\to 5}{\frac {2}{x-5}}}.
Include values greater than and less than C.{\ display style c} function. If you don’t have a graph of the function, take the values to the right and left of c{\displaystyle c} to see if the limits are different on each side of the function. If they are, you know that the limit cannot exist for the function.[7]
- For example, evaluate limx→0x|x|{\displaystyle \lim _{x\to 0}{\frac {x}{|x|}}}.
- Look at the left-hand side of the function or limx→0−x|x|{\displaystyle \lim _{x\to 0-}{\frac {x}{|x|}}}. Enter a value to the left of 0 or less than 0, such as -1: −1|−1|=−11=−1{\displaystyle {\frac {-1}{|-1|}} ={\ fraction {-1}{1}}=-1}.
- Look at the right hand side of the function or limx→0+x|x|{\displaystyle \lim _{x\to 0+}{\frac {x}{|x|}}}. Enter a value to the right of 0 or greater than 0, such as 1: 1|1|=11=1{\displaystyle {\frac {1}{|1|}}={\frac {1} {1} } =1}.
- Limits to the left and right of limx→0x|x|{\displaystyle \lim _{x\to 0}{\frac {x}{|x|}}} they are different, so there is no limit.
- If you have a calculator with you, include several different values greater than and less than x{\displaystyle x} which are closer to c{\displaystyle c}. For the example above, you can include -0.9, -0.5, 0.5, and 0.9.
- For example, evaluate limx→0x|x|{\displaystyle \lim _{x\to 0}{\frac {x}{|x|}}}.
Find the limit using algebra. Instead of using a graph to understand how a function behaves around a limit, use your understanding of algebra. Knowing that a square root can never be negative or that you can’t divide it by 0 helps you determine if a function is defined as x{\displaystyle x} approaches some value c{\displaystyle c}. If not, you know there is no limit.[8]
- For example, calculate limx→−2−2x+2{\displaystyle \lim _{x\to -2}{\frac {-2}{x+2}}}.
- If you include -2 in the function, the denominator equals 0: −2−2+2=−20{\displaystyle {\frac {-2}{-2+2}}={\frac {-2} { 0 }} }.
- It cannot be divided by 0, and the function cannot be further simplified or factored to solve it. But you know that when x>−2{\displaystyle x>-2} -2}”> in the equation, the result becomes larger and more negative, so that it approaches negative infinity. When x<−2{\displaystyle x<-2}"https://wikimedia.org/api/rest_v1/media/math/render/svg/42512f84597c29ee0a1692b6952d5c4a6ad4620e" class="mwe-math-fallback-image-online" aria-hidden="TRUE" style="vertical alignment: -0.505ex; width: 7.399ex; height: 2.343ex;" alt="{\ display style xthe result becomes larger and more positive, so that it approaches positive infinity.
- Since the left and right sides approach infinity, there is no limit.
- For example, calculate limx→−2−2x+2{\displaystyle \lim _{x\to -2}{\frac {-2}{x+2}}}.
.
. The left side goes toward negative infinity as it gets closer, while the right side goes toward positive infinity. So there is no limit.
.
. Enter a value to the left of 0 or less than 0, such as -1: −1|−1|=−11=−1{\displaystyle {\frac {-1}{|-1|}} ={\ fraction {-1}{1}}=-1}
.
. Enter a value to the right of 0 or greater than 0, such as 1: 1|1|=11=1{\displaystyle {\frac {1}{|1|}}={\frac {1} {1} } =1}
.
.
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-2}”> in the equation, the result becomes larger and more negative, so that it approaches negative infinity. When x<−2{\displaystyle x<-2}"https://wikimedia.org/api/rest_v1/media/math/render/svg/42512f84597c29ee0a1692b6952d5c4a6ad4620e" class="mwe-math-fallback-image-online" aria-hidden="TRUE" style="vertical alignment: -0.505ex; width: 7.399ex; height: 2.343ex;" alt="{\ display style xthe result becomes larger and more positive, so that it approaches positive infinity.Categories: How to
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