
Consider the constant function \[f(x)=3\]. Let us try to find its limit as \[x=2\].
Answer
512.1k+ views
Hint: The limit of a function exists only if left hand limit and right hand limit exist and both are equal.
Also, the value of the limit will be equal to the value of the right hand limit and (or) the left hand limit.
We know, the limit of a function exists only if the left hand limit and right hand limit exist and both are equal.
So, first, we will find the left hand limit of the function \[f(x)=3\] at \[x=2\].
We know, the left hand limit of a function \[f(x)\] at \[x=a\] is given as
\[L.H.L=\underset{h\to 0}{\mathop{\lim }}\,f\left( a-h \right)\]
So, left hand limit of the function \[f(x)=3\] at \[x=2\] is given as
\[L.H.L=\underset{h\to 0}{\mathop{\lim }}\,f\left( 2-h \right)=3\][ \[\because \]\[f(x)\] is a constant function, value of a function is 3 for all \[x\in R\]]
Now , we will find the right hand limit of the function \[f(x)=3\] at \[x=2\].
We know the right hand limit of a function \[f(x)\] at \[x=a\] is given as
\[L.H.L=\underset{h\to 0}{\mathop{\lim }}\,f\left( a+h \right)\]
So , right hand limit of the function \[f(x)=3\] at \[x=2\] is given as
\[L.H.L=\underset{h\to 0}{\mathop{\lim }}\,f\left( 2+h \right)=3\]
We can clearly see that both the left hand limit and the right hand limit of the function \[f(x)=3\] exist at \[x=2\].
Also, the left hand limit and the right hand limit of the function \[f(x)=3\] at \[x=2\] are equal.
Since, the value of \[L.H.L=R.H.L\]at \[x=2\], hence, limit of the function \[f(x)=3\] exists at \[x=2\] and the value of limit of the function \[f(x)=3\] at \[x=2\] is \[3\].
Note: The graph of the given function \[f(x)=3\] is a straight line parallel to x-axis as shown in the figure.
From the graph, we can clearly see that whether we approach \[x=2\] from the left or right side, the value of the function is equal to \[3\].
So, the value of left hand limit of the function \[f(x)=3\] at \[x=2\] is equal to \[3\] and the value of right hand limit of the function \[f(x)=3\] at \[x=2\] is equal to \[3\].
Hence, the value of \[\underset{x\to 2}{\mathop{\lim }}\,f\left( x \right)\] is equal to \[3\].
Also, the value of the limit will be equal to the value of the right hand limit and (or) the left hand limit.
We know, the limit of a function exists only if the left hand limit and right hand limit exist and both are equal.
So, first, we will find the left hand limit of the function \[f(x)=3\] at \[x=2\].
We know, the left hand limit of a function \[f(x)\] at \[x=a\] is given as
\[L.H.L=\underset{h\to 0}{\mathop{\lim }}\,f\left( a-h \right)\]
So, left hand limit of the function \[f(x)=3\] at \[x=2\] is given as
\[L.H.L=\underset{h\to 0}{\mathop{\lim }}\,f\left( 2-h \right)=3\][ \[\because \]\[f(x)\] is a constant function, value of a function is 3 for all \[x\in R\]]
Now , we will find the right hand limit of the function \[f(x)=3\] at \[x=2\].
We know the right hand limit of a function \[f(x)\] at \[x=a\] is given as
\[L.H.L=\underset{h\to 0}{\mathop{\lim }}\,f\left( a+h \right)\]
So , right hand limit of the function \[f(x)=3\] at \[x=2\] is given as
\[L.H.L=\underset{h\to 0}{\mathop{\lim }}\,f\left( 2+h \right)=3\]
We can clearly see that both the left hand limit and the right hand limit of the function \[f(x)=3\] exist at \[x=2\].
Also, the left hand limit and the right hand limit of the function \[f(x)=3\] at \[x=2\] are equal.
Since, the value of \[L.H.L=R.H.L\]at \[x=2\], hence, limit of the function \[f(x)=3\] exists at \[x=2\] and the value of limit of the function \[f(x)=3\] at \[x=2\] is \[3\].
Note: The graph of the given function \[f(x)=3\] is a straight line parallel to x-axis as shown in the figure.

From the graph, we can clearly see that whether we approach \[x=2\] from the left or right side, the value of the function is equal to \[3\].
So, the value of left hand limit of the function \[f(x)=3\] at \[x=2\] is equal to \[3\] and the value of right hand limit of the function \[f(x)=3\] at \[x=2\] is equal to \[3\].
Hence, the value of \[\underset{x\to 2}{\mathop{\lim }}\,f\left( x \right)\] is equal to \[3\].
Recently Updated Pages
Master Class 11 Accountancy: Engaging Questions & Answers for Success

Glucose when reduced with HI and red Phosphorus gives class 11 chemistry CBSE

The highest possible oxidation states of Uranium and class 11 chemistry CBSE

Find the value of x if the mode of the following data class 11 maths CBSE

Which of the following can be used in the Friedel Crafts class 11 chemistry CBSE

A sphere of mass 40 kg is attracted by a second sphere class 11 physics CBSE

Trending doubts
Define least count of vernier callipers How do you class 11 physics CBSE

The combining capacity of an element is known as i class 11 chemistry CBSE

Proton was discovered by A Thomson B Rutherford C Chadwick class 11 chemistry CBSE

Find the image of the point 38 about the line x+3y class 11 maths CBSE

Can anyone list 10 advantages and disadvantages of friction

Distinguish between Mitosis and Meiosis class 11 biology CBSE
