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प्रश्न
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
उत्तर
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
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