Question

Discuss the continuity of the following function at the point(s) or in the interval indicated against them:

`f(x) = (5^x - e^x)/(2x)`  for x ≠ 0

= `1/2`(log5−1)         for x = 0 at x = 0

Answer 1

f(0) = `1/2`(log 5−1)  …[given]

`lim_(x→0) "f"(x) = lim_(x→0) (5^x - e^x)/(2x)`

= `lim_(x→0) (5^x - 1 - e^x + 1)/(2x)`

= `1/2 lim_(x→0) ((5^x - 1) - (e^x - 1))/x`

= `1/2 lim_(x→0) [((5^x - 1))/x - ((e^x - 1))/x]`

= `1/2(lim_(x→0) (5^x - 1)/x - lim_{x→0} (e^x - 1)/x)`

= `1/2`(log 5 - log e)   .... `[lim_(x→0) ("a"^x - 1)/x = log "a"]`

= `1/2`(log 5 - 1)   ...[∵ log e = 1]

∴ `lim_(x→0) "f"(x)` = f(0)

∴ f is continuous at x = 0