Limits And Continuity

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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\lim_{x\to a }  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 \lim_{x\to a^- } 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 \lim_{x\to a^+}f(x)It is the expected or estimated value of f(x) at xa 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  \lim_{x\rightarrow a^-}  f(x) = \lim_{x\rightarrow a^+}  f(x) = l.(say) 

Then, \lim_{x\rightarrow a}  f(x) exists & \lim_{x\rightarrow a}  f(x) = l.

We read it as the limit of f(x) as x approaches to a is equal to l.

Check Point

  1. Find \lim_{x\to0} f(x) when f(x)  x2 + 2x + 3.
  2. Find {\lim_{x\to1} f(x) when f(x) = \frac{x-1}{x^2+7x+5} . 
  3. Find {\lim_{x\to2} f(x) when f(x) = \frac{x^4-16}{x-2}
  4. Check the continuity of the function f(x) = 3x2 + 5x at x = 2.
  5. Find the value of a if f(x) = 3x2 + a is continuous at x = 2 and f(2) = 17.
Answer Key
  1. \lim_{x\to0}  f(x) = 3
  2. {\lim_{x\to1} f(x) = 0
  3. {\lim_{x\to2} f(x) = 32
  4. f(x) = 3x2 + 5x is continuous at x = 2.
  5. a = 5

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