Calculus > Limit Definition
The limit is a calculus term meaning the value that a given function will approach as a variable in the function moves towards some number. Its use is extensive in calculus, and has many real world applications. Taking the below function, you would read it as “The limit of a function of x as x approaches y is z.”
Squeeze Theorem and the Limit Definition
A simple illustration of the concept is to take the limit of the function y=x+2 as x approaches 5. In other words, we want to know what value of y the function approaches when x is 5. The variable x will need to approach 5 from both sides, unless otherwise noted (because there are two infinities: negative and positive).
Step 1:Take three steadily nearing values for x from both sides. For example, if you are asked to take the limit of x=5, you might choose the numbers 4.7, 4.9. 4.99. 5.01, 5.1 and 5.3. Try to choose numbers that are close to x.
Step 2:Solve for x-values. In this example:
When x = 4.7|4.9|4.99| 5 |5.01|5.1|5.3
f(x) = 6.7|6.9|6.99| 7 |7.01|7.1|7.3
Step 3:Note where the value is trending towards. In this example, y is trending towards 7.
This methodology is known as the Squeeze Theorem, or sandwich theorem, which describes the way in which you slowly “squeeze” the value of x towards the intended value. When shown this way, there is an obvious trend towards a value of 7 as x becomes closer to 5 from both directions. Therefore, the limit of the function is 7, which might seem like useless additional work given that simply taking the function at x=5 gives the same value.
A more Advanced Illustration of the Limit Definition
To find a case where a limit can offer a solution where simpler mathematics don’t quite work, take the function (x^2-1)/(x-1). Let’s say we want to find the limit at x=1. If you were to try and make x equal to 1, the function would be undefined due to division by 0. If you repeat the same method followed above, you will get the following results:
When x = 0.7|0.9|0.99| 1 |1.01|1.1|1.3
f(x) = 1.7|1.9|1.990|Undef|2.01|2.1|2.3
The value of f(x) as x approaches 1 becomes closer and closer to 2. This is enough proof for an answer.
Limits Near Infinity
A typical usage of limits is to find a value for numbers that trend towards infinity. For a function of (4x-2)/(x) there would be an undefined value at x = infinity. If you take arbitrarily increasing numbers, you can try to find a pattern to get the limit of the function as follows:
When x = 10000|1000000 |1000000000
f(x) = 3.9998|3.999998|3.999999998
As x approaches infinity, the value of the function steadily moves towards a value of four.
Advanced Techniques in Taking Limits
One of the advanced tricks in finding limits approaching infinity is to discard insignificant figures in the equation. In the equation (4x-2)/(x), we can discard the -2, since infinity outweighs -2 substantially. This leads you to 4x/x, which is 4. While not typically enough proof for a limit, the tip can help you tell if your derived limit is plausible or not.
Kouba, Duane. “Limits of Functions Using the Squeeze Principle.” http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/squeezedirectory/SqueezePrinciple.html