Continuity of a Function Example 3

Practice Questions — Continuity at a Point

Question 1.
Check the continuity of
$ \displaystyle f(x)=\left\{ {\begin{array}{*{20}{c}} {2x+3,} & {x<0} \\ {0,} & {x=0} \\ {{{x}^{2}}+3,} & {x>0} \end{array}} \right.$
at (x=0).

Step-by-Step Solution:

1. Left-hand limit at (x=0):
   [\lim_{x\to 0^-}(2x+3)][=2(0)+3=3]
2. Right-hand limit at (x=0):
   [\lim_{x\to 0^+}(x^2+3)][=0^2+3=3]
3. Value of function at (x=0):
   [
   f(0)=0
   ]

Conclusion:
[\text{LHL}] [= \text{RHL}] [= 3 \ne f(0)=0]
⇒ Function is discontinuous at (x=0) (removable discontinuity).

Question 2.
Check the continuity of
$ \displaystyle f(x)=\left\{ {\begin{array}{*{20}{c}} {{{x}^{2}}-4,} & {x\ne 2} \\ {0,} & {x=2} \end{array}} \right.$
at (x=2).

Step-by-Step Solution:

1. [\lim_{x\to 2}(x^2-4)][=4-4=0]
2. [
   f(2)=0
   ]

Conclusion:
[
\lim_{x\to 2}f(x)=f(2)=0
]
⇒ Function is continuous at (x=2).

Question 3.
Check continuity of
$ \displaystyle f(x)=\left\{ {\begin{array}{*{20}{c}} {3x-5,} & {x\le 1} \\ {{{x}^{2}}-1,} & {x>1} \end{array}} \right.$
at (x=1).

Step-by-Step Solution:

1. LHL:
   [\lim_{x\to 1^-}(3x-5)][=3(1)-5=-2]
2. RHL:
   [\lim_{x\to 1^+}(x^2-1)][=1^2-1=0]
3. Function value:
   [
   f(1)=3(1)-5=-2
   ]

Conclusion:
LHL = –2, RHL = 0 ⇒ Not equal
⇒ Discontinuous at (x=1) (jump discontinuity).

Question 4.
Check continuity of
[f(x)=\dfrac{x^2-9}{x-3}]
at (x=3).

Step-by-Step Solution:

1. Simplify:
   [\dfrac{x^2-9}{x-3}][=\dfrac{(x-3)(x+3)}{x-3}][=x+3,][\quad] [x\neq 3]
2. Limit:
   [
   \lim_{x\to 3}(x+3)=6
   ]
3. Function value:
   Undefined at (x=3)

Conclusion:
[\text{Limit exists but } f(3)][\text{ not defined}]
⇒ Discontinuous at (x=3) (removable discontinuity).

Question 5.
Check continuity of
[
f(x)=|x-2|
]
at (x=2).

Step-by-Step Solution:

1. LHL:
   [\lim_{x\to 2^-}|x-2|][=2-2=0]
2. RHL:
   [\lim_{x\to 2^+}|x-2|][=2-2=0]
3. Function value:
   [
   f(2)=0
   ]

Conclusion:
[\text{LHL}=\text{RHL}][=f(2)=0]
⇒ Function is continuous at (x=2).