Limits And Continuity
Calculus Worksheets
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.
Standard Limits
- Limit of polynomial function, f(x) = a0 + a1x + a2x2 + a3x3 + ….+ anxn
f(x)= f(a)
= (a0 + a1x + a2x2 + a3x3 + …. + anxn)
= (a0 + a1a + a2a2 + a3a3 + …. + anan)
- Limit of Rational function
=
=
provided g(x)
0.
=n
Algebra of limits of functions
If f(x) and g(x) be any two functions such that f(x) &
g(x) both exist.
[f(x)+g(x)] =
f(x) +
g(x)
[f(x) – g(x)] =
f(x) –
g(x)
[f(x) g(x)] =
f(x)
g(x)
=
where g(x)
0 .
Examples
Now let’s consider some examples on limits and continuity.
Example 1: Findf(x) where f(x) = x11 + 3x.
f(x) =
(x11 + 3x) = (1)11 + 3(1) = 4
Example 2: Find f(x) where f(x) =
.
f(x) =
=
= =
Example 3: Find f(x) where f(x) =
.
f(x) =
= 10
= 10 using
= n
Continuity
A function f(x) is called continuous at a point x = a in its domain if f(x)= f(a), which can also be stated as
f(x) =
f(x) = f(a).
Hence, if the left hand limit & right hand limit both exist, and are both equal to the value of f(x) at x = a, then the function f(x) is called continuous at a point x = a.
Example 4: Check the continuity of the function f(x) = x11 + 3x at x = 1.
f(x) =
= (1)11 + 3(1) = 4
f(1) = (1)11 + 3(1) = 4
Therefore, f(x) = f(1).
Hence, the function f(x) = x11 + 3x is continuous at x = 1.
Example 5: Find the value of a if f(x) = 2x + a is continuous at x = 1 and f(1) = 5.
Since f(x) = 2x + a is continuous at x = 1,
∴ f(x)=f(1)
(2x+a)=5
2(1)+a=5
a=5-2=3
Check Point
- Find
f(x) when f(x) = x2 + 2x + 3.
- Find
f(x) when f(x) =
.
- Find
f(x) when f(x) =
- Check the continuity of the function f(x) = 3x2 + 5x at x = 2.
- Find the value of a if f(x) = 3x2 + a is continuous at x = 2 and f(2) = 17.
Answer Key
f(x) = 3
f(x) = 0
f(x) = 32
- f(x) = 3x2 + 5x is continuous at x = 2.
- a = 5
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