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Significant Concepts
Significant Concepts
Formula
Formula
This is the basic formula for two sided limits.(one sided limits are written similarly, however a is either a+(right) or a-(left), this show how x either approaches a from the left or the right) B is the limit of f(x) as x approaches a. But what does that mean?
- a is the real number that x will approach
- f(x) is the function that we are analyzing the limit of
- b is the limit of the function
Example: lim x-->∞ 2x+4 = ∞
This means that we are looking at the output of the function 2x+4 as x approaches infinity. In this case, as x gets bigger and bigger the output also grows and does not reach a ceiling.
Dominant Terms
Dominant Terms
One important concept is the dominant term. The dominant term is whichever term has the largest exponent, and therefore has the greatest effect on the output. For example, if f(x)=x5+2x3+7x, x5 would be the dominant term because it has the greatest exponent. Also if a term has a variable exponent, like 5x for example, it will become the dominant term because as x grows, the exponent gets infinitely larger. This concept is very useful when working with fractions. If the dominant term is in the numerator, we know that the numerator will end up being larger than the denominator. For example, in (x4+x)/(87x3), as x approaches infinity so does y. If we took the reciprocal of that fraction so that the dominant term was in the denominator, it would approach 0. This is because as we divide by a bigger and bigger number, our output gets smaller and closer to 0.
Discontinuities
Discontinuities
A discontinuity creates a specific instance of two limit finite limits. Discontinuities can occur at a hole, where the two parts are barely separated, or there can be a larger gap in some cases. These large gaps can occur at asymptotes, where one side goes down and one goes up but they both approach the asymptote.
End behavior
End behavior
Another important concept is the end behavior of a function and how it is defined by a specific limit. When a is equal to positive or negative infinity, the limit defines the end behavior of a function.
Asymptotes
Asymptotes
There are many different types of asymptotes. One type of asymptote is the vertical asymptote. It corresponds to the zeros of the denominator of a rational function. That means that it would be located at x=(zero of the denominator). Another type of asymptote is the horizontal asymptote. In addition to these, there are also slant and polynomial asymptotes. One can find a horizontal asymptote on a graph if the degree of largest term of the numerator in the rational function, is less than or equal to the degree of the largest term in the denominator. If that degree of the numerator is indeed less than the degree of the denominator then the horizontal asymptote is at y=0. If the degree of the numerator is equal to the degree of the denominator then the horizontal asymptote is at y= (leading coefficient of numerator)/(leading coefficient of denominator). If the degree of the numerator is however, one degree greater than that of the denominator, the slant asymptote(linear equation) is at y=((numerator)/(denominator))x. Polynomial asymptotes work in a similar way. If and only if the degree of the numerator is greater than one more than the degree of the denominator, then one may find a polynomial asymptote at y=((numerator)/denominator))x^(number the degree of the numerator exceeds the degree of the denominator). One observation a person might have at this point is that a slant asymptote is really just a polynomial asymptote with a degree of one because that is the number of degrees that the numerator exceeds the denominator by. That person would be right.
Application and Techniques
Application and Techniques
Tables
Tables
Tables can be very helpful tools in limit analysis. They make it easy to see the x and y values and how they relate to each other.
For example, if we are examining the limit of f(x)=x+2, we could make a table like this:
x y
1 3
2 4
3 5
This makes it easy to test different numbers and see what the output is, and it makes it clear which way the function is trending. For this example, as x approaches infinity y does too. It is also linear so it goes up by a common difference every time x increases. In this example, the slope it just 1.
Tables are also a great way to identify where a hole is. It helps to organize all your thoughts into one place. Take (x^2+5x+6)/(x+2).
x y
0 3
1 4
-1 2
2 5
-2 undefined
When we plug in -2, it is clear that there is a hole at -2. It is still possible to find this without a table, but a table helps to organize everything and makes it easier to visualize the equation. Using this chart, it makes it easy to see that this equation is a line. Without the table, you can still factor to (x+3)(x+2)/(x+2), which leaves our line as x+3 with a hole at -2. It comes down to preference if you like tables or not, but they are always a great place to start if you get lost.
Graphs
Graphs
If given a graph one would look for the point a line/curve approaches, but never reaches. If this limit is on the x-axis, the graph will approach infinity with the y-values, but will approach a real number with its x-values. If you already know the function, and what either x, or y approaches, then you can use a table. In this example as x approaches 2 y approaches 10, this is done using the first x-value as a base and then going slightly to right and left of the value on the number line, while trying to be as close as possible to the original number. A one sided limit is a limit that is different depending on the direction. As x approaches a value, f(x) would approach two different values, or as f(x) approaches a value x would approach two different values.
Finding the Dominant Term
Finding the Dominant Term
We know that the dominant term is the term with the greatest exponent. But what if there are multiple with the same exponent? This can be the case in fractions, where the numerator and denominator have equal dominant terms. To solve this, we need to use algebra and rearrange a little. For example, f(x) = (2x^3 - x^2)/(x^3 + x). Both the numerator and denominator have a x^3 term. Factoring out x^2 will leave (2x - 1)/(x + (3/x)). This can go further by factoring out another x, which leaves (2 - (1/x)/(1 + (3/x^2). Now it is clear where the dominant term is. In this case, the dominant term will make the denominator larger and larger as x approaches infinity.
Sample Problems
Sample Problems
1. f(x) = 2x + 6
lim(x→2) f(x) = ?
2. a(x) = -4x^6 + 1000x + 17469.37373
lim(x→∞) a(x) = ?
3. g(x) = (5x^4 - 12x + 10)/(18x + 15x^4 - 3x^3)
lim(x→∞) g(x) = ?
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