Select Page

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\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.

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

Give Your Child The eTutorWorld Advantage

Research has proven that personal online tutoring not just cements school learning, it helps build student confidence. Come to eTutorWorld for Expert Tutors and best K-12 Online Tutoring Services in the comfort and safety of your home at an affordable cost. Find a tutor online for Grade 3-12 Math, Science and English subjects and AP, SAT, SSAT and SCAT Test Prep help and test practice. Get free printable math and science worksheets in pdf format and SCAT Practice Tests. Sign up for a Free Trial Lesson Today!

© 2018 eTutorWorld - Online Tutoring and Test Prep | All rights reserved