NIMCET 2021 — Mathematics PYQ

To check the continuity of $f(x)$, we must examine the points where the definition changes, specifically at $x=-2$ and $x=-1$.

1. Simplifying the function for $x\in R-\{-1,-2\}$

The quadratic in the denominator can be factored:

So, for $x\ne -1,-2$:

2. Checking continuity at $x=-2$

• Function value: $f(-2)=-1$ (given).

• Limit value:

  $$\underset{x\to -2}{\lim }f(x)=\underset{x\to -2}{\lim }\frac{1}{x+1}=\frac{1}{-2+1}=-1$$

  Since ${\lim }_{x\to -2}f(x)=f(-2)$, the function is continuous at $x=-2$.

3. Checking continuity at $x=-1$

• Function value: $f(-1)=0$ (given).

• Limit value:

  $$\underset{x\to -1}{\lim }f(x)=\underset{x\to -1}{\lim }\frac{1}{x+1}$$

  As $x\to -1$, the denominator $x+1\to 0$, which means the limit approaches $\pm \infty$.

  Since the limit does not exist (it is an infinite discontinuity), the function is not continuous at $x=-1$.

4. Conclusion

The function is continuous everywhere except at $x=-1$. Therefore, the set of points where $f(x)$ is continuous is $R-\{-1\}$.

Final Answer

The function is continuous on the set:

The correct option is 2.