Limits And Continuity
Have you ever driven a motorbike or a car?
What do we observe when we press the accelerator for any one of them?
The motorbike/car attains its maximum speed after some time int`erval say 10-15 seconds.
Now, the speed might increase from 0 km/hr initially i.e. at time 0 seconds to 100/150 km/hr after time 10-15 seconds. Now, if we are interested in finding the approximate speed of the motorbike/car at the time say 8 seconds, then we want an expected or estimated value of the speed at a particular time instant.
Here, comes the role of limits.
Similarly, if we are not able to find or determine the actual value of a function say f(x) at any given point say x = a, we try to estimate i.e. find its expected value at x = a. This is where the concept of limit comes in picture.
Limit of a Function
Let f(x) be function and x = a be any point in its domain then the limit of f(x) at x = a is denoted by f(x) . It is the expected or estimated value of f(x) as x approaches to a.
- Left Hand limit: The left hand limit of f(x) at x = a is denoted by f(x). It is the expected or estimated value of f(x) at x = a when the values of f(x) near to and to the left of a are given.
- Right Hand limit: The right hand limit of f(x) at x = a is denoted by f(x). It is the expected or estimated value of f(x) at x= a when the values of f(x) near to and to the right of a are given
- Existence of limit: If the left hand limit & right hand limit both coincide i.e. are equal then the limit exists & the common value is called the limit of the function.
If f(x) = f(x) = l.(say)
Then, f(x) exists & f(x) = l.
We read it as the limit of f(x) as x approaches to a is equal to l.
Answer Key
- f(x) = 3
- f(x) = 0
- f(x) = 32
- f(x) = 3x^{2} + 5x is continuous at x = 2.
- a = 5
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