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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.

 

 

 Standard Limits

  1.  Limit of polynomial function, f(x) = a0 + a1x + a2x2 + a3x3 + ….+ anxn

 

\lim_{x\rightarrow a} f(x)= f(a)

 

\lim_{x\rightarrow a} (a0 + a1x + a2x2 + a3x3 + …. + anxn)

 

= (a0 + a1a + a2a2 + a3a3 + …. + anan)

 

  1. Limit of Rational function \lim_{x\rightarrow a} \left ( \frac{f(x)}{g(x)} \right ) = \frac{\left ( \lim_{x\rightarrow a} f(x) \right )}{\left ( \lim_{x\rightarrow a} g(x) \right )} = \frac{f(a)}{g(a)} provided g(x)\neq0.
  2. \lim_{x\rightarrow a} \left ( \frac{x^n-a^n}{x-a} \right ) =na^{n-1}

 

 

 

Algebra of limits of functions

 

If f(x) and g(x) be any two functions such that  \lim_{x\rightarrow a} f(x) & \lim_{x\rightarrow a} g(x) both exist.

 

  1. \lim_{x\rightarrow a} [f(x)+g(x)] = \lim_{x\rightarrow a}  f(x) + \lim_{x\rightarrow a}  g(x)
  2. \lim_{x\rightarrow a} [f(x) – g(x)] = \lim_{x\rightarrow a}  f(x) – \lim_{x\rightarrow a} g(x)
  3. \lim_{x\rightarrow a} [f(x) g(x)] = \lim_{x\rightarrow a}  f(x)\lim_{x\rightarrow a} g(x)
  4. \lim_{x\rightarrow a} \left [ \frac{f(x)}{g(x)} \right ] =  \frac{\lim_{x\to a}f(x) }{\lim_{x\to a}g(x) } where g(x) \neq 0 .

 Examples

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

 

Example 1: Find{\lim_{x\to1}f(x) where f(x) = x11 + 3x.

 

{\lim_{x\to1}f(x) = {\lim_{x\to1}(x11 + 3x) = (1)11 + 3(1) = 4

 

 Example 2: Find {\lim_{x\to2}f(x) where f(x) = \frac{x+1}{x^2+2x+3} .

 

{\lim_{x\to2}f(x) ={\lim_{x\to2} \left ( \frac{x+1}{x^2+2x+3} \right ) = \frac{\lim_{x\to2}(x+1) }{\lim_{x\to2} (x^2+2x+3)}

 

\frac{(2+1) }{(2)^2+2(2)+3} = \frac{3}{11}

  

Example 3: Find {\lim_{x\to1} f(x) where f(x) = \frac{x^{10}-1}{x-1} .

 

{\lim_{x\to1} f(x) = {\lim_{x\to1}\left ( \frac{x^{10}-1^{10}}{x-1} \right ) = 10(1)^{10-1} = 10 using \lim_{x\rightarrow a} \left ( \frac{x^n-a^n}{x-a} \right ) = na^{n-1}

  

Continuity

 

A function f(xis called continuous at a point x = a in its domain if\lim_{x\rightarrow a}  f(x)f(a), which can also be stated as

 

\lim_{x\to a^+}f(x) =\lim_{x\to a^-}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.

 

{\lim_{x\to1} f(x) = {\lim_{x\to1} (x^{11}+3x)= (1)11 + 3(1) = 4

 

f(1) = (1)11 + 3(1) = 4

 

Therefore, {\lim_{x\to1} 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,

 

{\lim_{x\to1} f(x)=f(1)

 

{\lim_{x\to1}(2x+a)=5 

 

2(1)+a=5

 

a=5-2=3

 

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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