Determine the value(s) of constant(s) involved in the definition so that the given function is continuous: `f(x)={(sin2x)/(5x)\ \ \ ,\ \ \ if\ x!=0 3k\ \ \ ,\ \ \ if\ x=0`

Explanation

`f(x)=frac{sin 2 x}{5 x}, text { if } x neq 0 k, text { if } x=0`


If


`f(x)` is continuous at `x=0`, then


`lim _{x rightarrow 0} f(x)=f(0)`


`lim _{x rightarrow 0} frac{sin 2 x}{5 x}=k`


`lim _{x rightarrow 0} frac{2 sin 2 x}{5 times 2 x}=k`


`frac{2}{5} lim _{x rightarrow 0} frac{sin 2 x}{2 x}=k`


`frac{2}{5} times 1=k`


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