Question

If f(x) = `{((3^x - e^x)/(tan2x);     x ≠ 0),(1/2(log3 + 1);    x = 0):}` then ______ 

Options

• f(x) is continuous at x = 0

• f{x) is discontinuous at x = 0

• `lim_{x→0}`f(x) does not exist

• none of these

Answer 1

If f(x) = `{((3^x - e^x)/(tan2x);     x ≠ 0),(1/2(log3 + 1);    x = 0):}` then f{x) is discontinuous at x = 0.

Explanation:

`lim_{x→0}f(x) = lim_{x→0} (3^x - e^x)/(tan2x)`

= `lim_{x→0} (3^x - 1 + 1 - e^x)/(tan 2x)`

= `lim_{x→0} ((3^x - 1)/x - (e^x - 1)/x)/((tan2x)/(2x) xx 2)`

= `(log3 - loge)/2`

= `1/2(log3 - 1)`

∴ `lim_{x→0}f(x) ≠ f(0)`

∴ f(x) is discontinuous at x = 0.