NIMCET 2020 — Mathematics PYQ

A function is continuous at $x=a$ if:

1. Find the Left-Hand Limit (LHL)

For x < 2, we use $f(x)=\frac{5}{2}-x$:

2. Find the Right-Hand Limit (RHL)

For x > 2, we use $f(x)=x-\frac{3}{2}$:

3. Find the Value of the Function at $x=2$

From the given definition, when $x=2$:

Conclusion

Comparing the results:

• $\text{LHL}=\frac{1}{2}$

• $\text{RHL}=\frac{1}{2}$

• $f(2)=1$

Since the $\text{LHL}=\text{RHL}$, the limit exists and is equal to $\frac{1}{2}$. However, the limit value ($\frac{1}{2}$) is not equal to the function value ($1$).

Therefore, the function is discontinuous at $x=2$. This is known as a removable discontinuity.

Correct Option: (B)