Calculus 1 problem: functions?

Im doing some older exams that my professor has provided, but I havent got the solutions for this one. Can someone help confirm that the solution Ive arrived at is correct?

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  • Ray S
    Lv 7
    1 month ago

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    A function ƒ is continuous at x=a if:

               𝟏.  lim ƒ(x)  exists

                    𝐱➔𝓪

               𝟐.  ƒ(𝓪)  exists

               𝟑.  lim ƒ(x) = ƒ(𝓪)

                    𝐱➔𝓪

    —————————————————————————————

    Since (ℯˣ -1) / x   is   0/0   at x = 0,   lim (ℯˣ -1) / x   as   x➔0   can be

    determined by using L'Hopital's Rule.

    Therefore, lim( (ℯˣ-1)/x ) = lim( ℯˣ/1 ) = ℯ⁰ = 1   

                       x➔0                   x➔0                           

    so that defining the point (0,1) fills the hole in the graph of

    y = (ℯˣ -1) / x  making g(x) continuous at x = 0.

    —————————————————————————————

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  • fcas80
    Lv 7
    1 month ago

    (e^x - 1)/x is undefined at x = 0.  It is a 0/0 expression.  But by L'Hospital's Rule, its limit is 1.  And further, the function g is defined to be 1, although this is not needed.  So yes, it is continuous.

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  • 1 month ago

    A is correct. See graph below.

    Attachment image
    • Robin1 month agoReport

      Thank you.

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