Limits at infinity

By limits at infinity we mean one of the following two limits.

In other words, we are going to be looking at what happens to a function if we let x get very large in either the positive or negative sense. Also, as we'll soon see, these limits may also have a value at infinity.

One of the mysteries of Mathematics seems to be the concept of infinity, usually denoted by the symbol ∞ . So what is ∞ ? It is simply a symbol that represents large numbers. Indeed, numbers are of three kinds: large, normal size, and small. The normal size numbers are the ones that we have a clear feeling for. For example, what does a trillion mean? That is a very large number. Also numbers involved in macro-physics are very large numbers. Small numbers are usually used in micro-physics. Numbers like 10^-75 are very small. Being positive or negative has special meaning depending on the problem at hand. The common mistake is to say that -∞ is smaller than 0. While this may be true according to the natural order on the real line in term of sizes, -∞ is big, very big!

whenever you take the inverse of small numbers, you generate large numbers and vice-versa. Mathematically we can write this as:

Do not treat +∞ and -∞ as ordinary numbers. These symbols do not obey the usual rules of arithmetic, for instance, ∞ + 1 = ∞, ∞ - 1= ∞, 2. ∞ = ∞ Here we're going to learn what is happening to the graph on an infinite interval. That often means, what is the y value approaching when x approaches infinity.

Definition of Limits at Infinity

Limits at infinity maybe defined as if the limit exists at infinity, the y value of a graph is always within certain units of y that exists as x approaches infinity. In other words, what is the value that y is approaching when x approaches infinity.

Let's say the limits at infinity for the graph is A, then y=A would be a horizontal asymptote for the graph.

The equation for Horizontal asymptote is y=limits at infinity.

• limits when x approaches infinity for function c over x to the r =0 when r is any positive ration number and c is a real number. Because as x approaches infinity, x to the r thus will become infinity too, and any number c, no matter how big c is over infinity is defined to be 0. (c /x^r)= 0 if r is positive rational number and c is a real number.

• If the degree of the numerator is less than the degree of the denominator then the limit of the rational function is 0.

  Example:  (2x+5)/(3x²+1)= 0 because the degree for 2x+5 is smaller than 3x²+1.

• If the degree of the numerator and the degree of the denominator are the same, then the limit of the rational function as it approaches infinity is the ratio of the leading coefficients.

  Example:  (x+1)/(x+2)= 1 because 1/1=1.

• If the degree of numerator is greater than degree of denominator, then the limit of the rational function does not exist, because it will be infinity.

  Example:  (x³+1)/(x²+2) does not exist because 3 > 2

• Graphically, if the graph seems to be approaching some y value but doesn't get there on the graph you see as x appraoches to infinity and negative infinity, then the y value would be its limit.