1.

If the function ` f(x) = (( 3x + 4 tan x))/x ` continuous at x=0? If not, hwo may the funcation be defined to make it continuous at this point ?

Answer» Since f(x) is not defined at x = 0 , it cannot be continuous at x=0
However, ` lim_(x to 0) f(x) = lim_(x to 0) (( 3x + 4 tan x)/x) = lim_( x to 0) [ 3+4 .( sin x)/ . 1/ (cos x) ]`
` = 3+4. lim_( x to 0) { ( sin x)/x } . { lim_( x to 0) 1/ ( cos x)} =7 `
So, in order to make f(x) continuous at x =0 , we define it as
`f(x)= {{:(((3x + 4 tanx))/(x)",","when", x ne0),(7",","when" , x=0):}`


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