Question

If $f\left(x\right)=\frac{\sin 4x}{5x}+a$ for $x>0$
$=x+4-b$ for $x<0$
$=1$ for $x=0$
is continuous at $x=0.$ Find $a$ & $b.$

Answer 1

$f\left(x\right)=\{\begin{array}{c}\frac{\sin 4x}{5x}+a,x>0\\ x+4-b,x<0\\ 1,x=0\end{array}$ is continuous at $x=0$

$\to$ Condition of continuity

${lim}_{x\to 0^{+}}\frac{\sin 4x}{5x}+a={lim}_{x\to 0^{-}}(x+4-b)=f\left(0\right)$

$\Rightarrow {lim}_{h\to 0}\frac{\sin 4(0+h)}{5(0+h)}+a={lim}_{h\to 0}(0-h+4-b)=1$

$\Rightarrow {lim}_{h\to 0}\frac{4}{5}\left(\frac{\sin 4h}{4h}\right)+a=4-b=1$

$\Rightarrow \frac{4}{5}+a=4-b=1$

$\Rightarrow b=3$

& $a=\frac{1}{5}$

$\Rightarrow b=3$ & $a=\frac{1}{5}$