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Question
Evaluate the following :
`lim_(x -> 1) [(4^(x - 1) - 2^x + 1)/(x - 1)^2]`
Solution
`lim_(x -> 1) [(4^(x - 1) - 2^x + 1)/(x - 1)^2]`
put x – 1 = h
∴ x = 1 + h
As x → 1, h → 0
Required limit
= `lim_("h" -> 0) (4^"h" - 2^(1 + "h") + 1)/("h"^2)`
= `lim_("h" -> 0) ((2^2)^"h" - 2^1 * 2^"h" + 1)/("h"^2)`
= `lim_("h" -> 0) ((2^"h")^2 - 2(2^"h") + 1)/("h"^2)`
= `lim_("h" -> 0) (2^"h" - 1)^2/("h"^2)`
= `lim_("h" -> 0) ((2^"h" - 1)/"h")^2`
= (log 2)2 ...`[because lim_(x -> 0) ("a"^x - 1)/x = log"a"]`
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