4 For what value of k is the following function continuous at x = 2 f(x)= {2x+1 - Maths - Determinants

Question

4 For what value of k is the following function continuous at x
= 2 ?
f(x)= {2x+1 if x less than 2
k if x=2
3x-1 if x is greater than 2

Shridhar Kaushik · Accepted Answer

We have,

$f (x) = {\begin{matrix} \begin{matrix} 2 x + 1 if x < 2 \\ k if x = 2 \end{matrix} \\ 3 x - 1 if x > 2 \end{matrix}$

Now,

$L.H.L. = \lim_{x \to 2^{-}} f (x) = \lim_{x \to 2^{-}} (2 x + 1) = \lim_{h \to 0} [2 (2 - h) + 1] = \lim_{h \to 0} [5 - 2 h] = 5 ...... (1)$

$and R.H.L. = \lim_{x \to 2^{+}} f (x) = \lim_{x \to 2^{+}} (3 x - 1) = \lim_{h \to 0} [3 (2 + h) - 1] = \lim_{h \to 0} [6 + 3 h - 1] = 5 + 0 = 5 ...... (2) and f (2) = k .... (3)$

We know that f(x) is continuous at x = a if

$L.H.L. = R.H.L. = f (a) \Rightarrow L.H.L. = R.H.L. = f (2) [∵ a = 2] \Rightarrow 5 = 5 = k \Rightarrow k = 5$

Thus, the value of k is 5.

4. For what value of k is the following function continuous at x = 2 ? f(x)= {2x+1 if x less than 2 k if x=2 3x-1 if x is greater than 2

4. For what value of k is the following function continuous at x = 2 ?

f(x)= {2x+1 if x less than 2

k if x=2

3x-1 if x is greater than 2