Write the value of b for which

Question:

Write the value of $b$ for which $f(x)=\left\{\begin{array}{rl}5 x-4 & 0

Solution:

Given: $f(x)=\left\{\begin{array}{l}5 x-4,0

If $f(x)$ is continuous at $x=1$, then

$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)=f(1)$                   ....(1)

Now,

$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{h \rightarrow 0} f(1-h)=\lim _{h \rightarrow 0} 5(1-h)-4=5-4=1$

$\lim _{x \rightarrow 1^{+}} f(x)=\lim _{h \rightarrow 0} f(1+h)=\lim _{h \rightarrow 0} 4(1+h)^{2}+3 b(1+h)=4+3 b$

Also, $f(1)=5(1)-4=1$

$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)=f(1) \quad$ [From eq. (1)]

$\Rightarrow 1=4+3 b=1$

$\Rightarrow 1=4+3 b$

$\Rightarrow-3=3 b$

$\Rightarrow b=-1$

Thus, for $b=-1$, the function $f(x)$ is continuous at $x=1$.

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