Graph of e x − 1 x i am trying to draw the graph of f ( x ) = e x − 1 x .its domain is R -0 but its limit is 1 in the neighbourhood of zero.Also l i m x → − ∞ f ( x ) = 0 so negative X axis is asymptote.also d f ( x ) d x = ( x − 1 ) e x + 1 x 2 so f(x) is increasing in ( 0 ∞ ) and decreasing in ( − ∞ 0 )now the ambiguity is f ( − ∞ ) → 0 and f ( 0 − ) → 1 but f(x) is decreasing in negative interval.

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Graph of                     e        x            −      1        x  i am trying to draw the graph of   f  (  x  )  =                    e        x            −      1        x  .its domain is       R  -0 but its limit is 1 in the neighbourhood of zero.Also   l  i      m          x      →      −      ∞        f  (  x  )  =  0 so negative X axis is asymptote.also             d      f      (      x      )              d      x        =            (      x      −      1      )              e        x            +      1              x      2      so f(x) is increasing in   (  0    ∞  ) and decreasing in   (  −  ∞    0  )now the ambiguity is   f  (  −  ∞  )  →  0 and   f  (      0    −    )  →  1 but f(x) is decreasing in negative interval.

Dobricap · Accepted Answer

Step 1There is no ambiguity. The mistake is when you claim that f is decreasing in   (  −  ∞  ,  0  ) since for   x  &amp;lt;  0 you still have       f    ′    (  x  )  =            (      x      −      1      )              e        x            +      1              x      2        &amp;gt;  0.Step 2This is because of the well-known       e    x    ≥  x  +  1 for all x, in particular   1  −  x  ≤      e          −      x       i.e.   (  1  −  x  )      e    x    ≤  1 and hence   (  x  −  1  )      e    x    +  1  ≥  0.So f is increasing everywhere which goes along with   f  (  −  ∞  )  &amp;lt;  f  (  0  )

Graph of (e^x-1)/x

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