Answer

Verified

433.2k+ views

**Hint**: In this question, we need to check the continuity of the given function at x=1 and x=-1. For this, we will use the relation of the left hand limit and the right hand limit for the given function.

**:**

__Complete step-by-step answer__For a function to be continuous, the functions should have the same value at the left-hand limit, right-hand limit as well as on the limits.

Here in the question, we need to determine the continuity at the point x=1 and x=-1 as the function given is associated with the same.

Now, the value of the function at x=-1 is defined as $ f\left( x \right) = - 2 $ so, the value of the function at x=-1 is given by

$ f( - 1) = - 2 $

The value of the left-hand limit of the function at x=-1 is defined as $ f\left( {{x^ - }} \right) = - 2 $ so, the value of the function at $ x = - 1 - h $ where h is infinitesimally small and is tending to zero is given by

$

f\left( {{x^ - }} \right) = \mathop {\lim }\limits_{x \to - {1^ - }} \left( { - 2} \right) \\

= \left( { - 2} \right) \\

= - 2 - - - - (i) \\

$

The value of the right-hand limit of the function at x=-1 is defined as $ f\left( {{x^ + }} \right) = 2x $ so, the value of the function at $ x = - 1 + h $ where h is infinitesimally small and is tending to zero is given by

\[

f\left( {{x^ + }} \right) = \mathop {\lim }\limits_{x \to - {1^ + }} 2x \\

= \mathop {\lim }\limits_{h \to 0} \left( {2( - 1 + h)} \right) \\

= \mathop {\lim }\limits_{h \to 0} \left( { - 2 + 2h} \right) \\

= - 2 - - - - (ii) \\

\]

From the equations (i) and (ii) we can see that the value of the functions at the left-hand limit is equal to the value of the function at the right-hand limits, so the given function is not continuous at x=-1.

Now, the value of the function at x=1 is defined as $ f\left( x \right) = 2x $ so, the value of the function at x=1 is given by

$

f(1) = 2x \\

= 2 \\

$

The value of the left-hand limit of the function at x=1 is defined as $ f\left( {{x^ - }} \right) = 2x $ so, the value of the function at $ x = 1 - h $ where h is infinitesimally small and is tending to zero is given by

$

f\left( {{x^ - }} \right) = \mathop {\lim }\limits_{x \to {1^ - }} 2x \\

= \mathop {\lim }\limits_{h \to 0} \left( {2(1 - h)} \right) \\

= \mathop {\lim }\limits_{h \to 0} \left( {2 - 2h} \right) \\

= 2 - - - - (iii) \\

$

The value of the right-hand limit of the function at x=1 is defined as $ f\left( {{x^ + }} \right) = 2 $ so, the value of the function at $ x = 1 + h $ where h is infinitesimally small and is tending to zero is given by

$

f\left( {{x^ + }} \right) = \mathop {\lim }\limits_{x \to {1^ + }} \left( 2 \right) \\

= 2 - - - - (iv) \\

$

From the equations (iii) and (iv) we can see that the value of the functions at the left-hand limit is equal to the value of the function at the right-hand limits, so the given function is not continuous at x=1.

Hence, the given function is continuous at x=-1 and x=1.

**Note**: If a function is differentiable, then, the function must be continuous at the point of the investigation while it is not necessary that if it is continuous then, it must be differentiable as well.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Give a reason for the establishment of the Mohammedan class 10 social science CBSE

What are the two main features of Himadri class 11 social science CBSE

The continent which does not touch the Mediterranean class 7 social science CBSE

India has form of democracy a Direct b Indirect c Presidential class 12 sst CBSE

which foreign country is closest to andaman islands class 10 social science CBSE

One cusec is equal to how many liters class 8 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which foreign country is closest to Andaman Islands class 11 social science CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE