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# Limits And Continuity

## Calculus Worksheets

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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 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 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 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

1.  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)

1. Limit of Rational function  = = provided g(x) 0.
2.  =n ### Algebra of limits of functions

If f(x) and g(x) be any two functions such that f(x) & g(x) both exist.

1. [f(x)+g(x)] = f(x) + g(x)
2. [f(x) – g(x)] = f(x) – g(x)
3. [f(x) g(x)] = f(x) g(x)
4.  = where g(x) 0 .

Examples

Now let’s consider some examples on limits and continuity.

Example 1: Find f(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(xis 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(xat x = a, then the function f(xis 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

1. Find f(x) when f(x)  x2 + 2x + 3.
2. Find f(x) when f(x) = .
3. Find f(x) when f(x) = 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.
1. f(x) = 3
2. f(x) = 0
3. f(x) = 32
4. f(x) = 3x2 + 5x is continuous at x = 2.
5. a = 5

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