Question

Consider the following:
1. $x+x^2$ is continuous at $x=0$
2. $x+\cos \frac{1}{x}$ is discontinuous at $x=0$
3. $x^2+\cos \frac{1}{x}$ is continuous at $x=0$
Which of the above are correct?

• A. $1$ and $2$ only
• B. $2$ and $3$ only
• C. $1$ and $3$ only
• D. $1,2$ and $3$

Answer 1

The correct option is A $1$ and $2$ only

$x+x^2$ is a polynomial function and hence, is continuous on $R$. So, statement 1 is true.

$x+\cos \frac{1}{x}$ has a polynomial part and a non-polynomial part. $x$ is a polynomial function and hence, is continuous on $R$.

But, for $\cos \frac{1}{x}$, $LHL=\underset{x\to 0^{-}}{lim}\cos \frac{1}{x}\to \cos -\infty \ne \underset{x\to 0^{+}}{lim}\cos \frac{1}{x}=\cos \infty$ where $\cos \infty$ is an unknown and hence, the function is discontinuous. So, statement 2 is true.

$x^2+\cos \frac{1}{x}$ has a polynomial part and a non-polynomial part. $x^2$ is a polynomial function and hence, is continuous on $R$. But, for $\cos \frac{1}{x}$, $LHL=\underset{x\to 0^{-}}{lim}\cos \frac{1}{x}\to \cos -\infty \ne \underset{x\to 0^{+}}{lim}\cos \frac{1}{x}=\cos \infty$ where $\cos \infty$ is an unknown and hence, the function is discontinuous. So, statement 3 is false.