Limits of Infinity Simplified

In Mathematics, one of the mysteries is the concept of ‘infinity’ which represents very large numbers.  We come across +infinity and –infinity. 1 divided by zero gives us + or – infinity. To be precise 1/0+ gives us +infinity and 1/0- gives us –infinity. But 1/infinity is undefined as we cannot determine the value of it.  Let us consider a variable x and arrive to the value of 1/x. When we plug in large values of x, the value of 1/x gets smaller and smaller. We can conclude that as x gets larger, 1/x tends to zero but does not equal zero.  To say that the value equals zero we use the term limit in mathematics, using which we can say, as the value of x approaches infinity, limit of 1/x equals zero. Here is where infinite limits come into picture, they are some functions which increase or decrease without bounds near certain values for the independent variable. Infinite limits are represented as , lim (x->-infinity) a/x^r= zero and lim(x->+infinity) a/x^r= zero for any number a, r being a positive integer and x^r is a defined function.  We also have lim(x->infinity)x^r= infinity and for any natural number ‘n’, lim(x->-infinity)x^n=(-1)^n infinity.

Let us take a look at some of the limits at infinity examples, evaluate lim(x->infinity) [x^2+1]/[2x-3]. Let us re-write the given function, (x^2/x).[1+1/x^2]/[2-3/x] which gives us x.[1+1/x^2]/[2-3/x]. Applying limit we get, lim(x->infinity)[1+1/x^2]/[2-3/x] gives us (1/2)x  and 1/2 lim(x->infinity)x= infinity

Finding infinite limits of a given function, infinite limits are given by lim(x->a) f(x) = +infinity or lim(x->a) f(x)= - infinity. Solving Infinite limits, for instance let us solve lim(x->0-)[x^2-1/x]. We can see this limit is going to infinity, here we need to determine whether it is positive or negative infinity. Simplifying [x^2-1/x] we get [x^3/x – 1/x] which gives (x^3 -1)/x. Now substituting (-0.1) for x. we get, [(-0.1)^3-1]/(-0.1)= negative/negative  which gives positive and hence we get positive infinity as the answer.

Infinite limits are defined as lim(x->a) f(x)= infinity  for all x close to x=a, from both sides, but not letting x=a and limit(x->a) f(x)= - infinity for all x close to x=a, from both sides, but not letting x=a. Evaluating infinite Limits, lim(x->0-)3/x^2. As we can see, it is a left hand limit. In this case, we take smaller and smaller values of x and when we square them, they will get smaller, but on squaring the values will be positive. So, now we have a constant divided by an increasingly small positive number which gives a positive infinity, that is lim(x->0-)3/x^2=positive infinity.