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Question
Evaluate the following Limits: `lim_(x -> 0) ("e"^x + e^(-x) - 2)/x^2`
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Solution
`lim_(x -> 0) ("e"^x + e^(-x) - 2)/x^2`
= `lim_(x -> 0) ("e"^x + 1/"e"^x - 2)/x^2`
= `lim_(x -> 0) (("e"^x)^2 + 1 - 2"e"^x)/(x^2*"e"^x`
= `lim_(x -> 0) ((e^x - 1)^2)/(x^2*"e"^x)`
= `lim_(x -> 0) [(("e"^x - 1)/x)^2 xx 1/"e"^x]`
= `lim_(x -> 0) (("e"^x - 1)/x)^2 xx 1/(lim_(x -> 0) "e"^x`
= `(1)^2 xx 1/"e"^0 ...[lim_(x -> 0) ("e"^x - 1)/x = 1]`
= `1 xx 1/1`
= 1
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