Sometimes we can't find the actual value of a function directly, but we can see what it should be as we get closer and closer to that particular value. A limit is a value that a function (or a sequence) "approaches" as the input (or index) "approaches" some value.
Let’s understand this with an example.
Consider the function x2 - 1/ x - 1
We know that the given function is not defined when the value of x is 1, because division by zero is not a valid mathematical operation. So, we can find value as x approaches 1.
Let’s try approaching when x tends to 1.
From the above example, we can see that when x gets close to 1, then the value of x2 - 1/ x - 1 gets close to 2.
When x = 1, we don’t know the value (as it is indeterminate form). We can see that value gets close to 2. We can give answers as 2 but it is not the actual value. Hence the concept of the word “limit” came into existence.
The value of x2 - 1/ x - 1 as x approaches 1 is 2.
Something that is boundless or endless or else something that is larger than any real number is known as infinity. It is denoted by a symbol ∞.
Let’s understand with an example.
Find the value of one divided by infinity (1/∞).
Infinity is not a defined value. So, 1/∞ is similar to 1/smart.
We could say that 1/∞ = 0, but how it is possible because if we try to divide 1 into infinite pieces they can end up to 0 each. You may think about what happened to 1.
In fact, the value of 1/∞ is known to be undefined.
So, instead of trying to find it for infinity. Let’s try it for a larger value.
From the above table, we can see that as x gets larger, 1/x tends to 0.
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We can conclude two fact:
Hence the limit of 1/x as x approaches infinity is 0. We can write it as
lim (1/x) = 0 when x approaching ∞.
In a mathematical way, we are not talking about when x = ∞, but we know the value as x gets bigger the value gets closer and closer to 0.
So, infinity can’t be used directly but we can use the limit.
Limits to Infinity:
How to find the limit of a function as x approaches infinity?
Let function be y = 2x.
So, from the above table, we can say that as x approaches infinity, then 2x also approaches infinity.
Don’t consider “=” sign as the exact value in the limit. We can’t actually get to infinity, but in limit language the limit is infinity.
Infinity and Degree
Functions like 1/x approaches to infinity. This is also valid for 1/x2 and so on.
A function such as x will approach infinity, same we can apply for 2x or x/9, and so on. Likewise functions with x2 or x3 etc will also approach infinity.
We should be careful with negative functions like -x will approach -infinity. So we have to look at the sign of x and then decide the function value.
Let’s understand negative function value with an example:
Consider 2x2 - 4x
Sol: We know that 2x2 will tend towards +infinity and -5x will tend towards -infinity. But x2 value will be larger as compared to x. So 2x2 - 4x will tend to +infinity.
When we look for the degree of the function, check the highest exponent in the function.
The degree of function is divided into two parts:
The degree is greater than 0, the limit is infinity.
The degree is less than 0, the limit is 0.
A rational function is one that is the ratio of two polynomial functions.
Let f(x) = P(x)/Q(x)
P(x) = x3 + 2x - 1 and Q(x) = 3x2
Now compare the degree of P(x) to the degree of Q(x).
If the degree of P is less than the degree of Q the limit value is 0.
If the degree of P and Q are the same divide the coefficient of terms with the largest exponent.
If the degree of P is greater than the degree of Q then two cases come.
The limit value is positive infinity.
Or maybe negative infinity. We need to look at the sign and then decide.