NIMCET 2020 — Mathematics PYQ

Concept:

Definition:

• A function f(x) is said to be continuous at a point x = a in its domain, if ${\lim }_{x\to a}f(x)$ exists or or if its graph is a single unbroken curve at that point.

• f(x) is continuous at x = a $\Rightarrow$ ${\lim }_{x\to a^{+}}f(x)={\lim }_{x\to a^{-}}f(x)={\lim }_{x\to a}f(x)=f(a)$.

Calculation:

For x $\ne$ 0, the given function can be re-written as:

$f(x)=\{\begin{array}{ll}\frac{1-x}{2},& amp;x\ne 0\\ K,& amp;x=0\end{array}$

Since the equation of the function is same for x < 0 and x > 0, we have:

${\lim }_{x\to 0^{+}}f(x)={\lim }_{x\to 0^{-}}f(x)={\lim }_{x\to 0}\frac{1-x}{2}=\frac{1-0}{2}=\frac{1}{2}$

For the function to be continuous at x = 0, we must have:

${\lim }_{x\to 0}f(x)=f(0)$

$\Rightarrow K=\frac{1}{2}$