Question

Let f(x) be a function defined by f(x)={4x−5, if 'x' is less than or equal to 2. {x- lambda, if 'x' is greater than 2. find lambda if lim x tends to 2 f(x) exists.

Answer 1

$\rm\large\underline{\text{Answer}}$

$\lambda=-1$

$\rm\large\underline{\text{Explanation}}$

Given function

$f(x)=\begin{cases} 4x-5 &(x \leq 2)\\ x-\lambda &(x>2)\end{cases}$

The LHL and RHL

$\boxed{\begin{aligned}\displaystyle&\lim_{x \to 2-}f(x)=\lim_{x \to 2-}(4x-5)=3\\\\&\lim_{x \to 2+}f(x)=\lim_{x \to 2+}(x-\lambda)=2-\lambda\end{aligned}}$

The limit exists if and only if LHL and RHL are equal.

$\cdots\longrightarrow 3=2-\lambda$

$\cdots\longrightarrow \lambda=-1$

So,

$\cdots\longrightarrow \boxed{\lambda=-1}$

$\rm\large\underline{\text{Extra explanation}}$

If the limit which equals the functional value exists, the function is called to be continuous.

$\boxed{\begin{aligned}\displaystyle\lim_{x \to a-}f(x)=f(a)=\lim_{x \to a+}f(x)\end{aligned}\text{ (Continuity)}}$

If the limit or functional value does not equal, the function is called to be discontinuous.

$\boxed{\begin{aligned}&\displaystyle\lim_{x \to a-}f(x)\neq\lim_{x \to a+}f(x)&\text{ (Discontinuity)}\\\\&\displaystyle\lim_{x \to a}f(x) \neq f(a)&\text{ (Discontinuity)}\end{aligned}}$