Calculate limit involving the antiderivative of a functionLet f : R → R , f ( x ) = e x 2 . Now, consider F an antiderivative of f.Calculate: lim x → ∞ x F ( x ) f ( x ) .I couldn't find the antiderivative of f, and besides finding F, I have no other ideas.

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Calculate limit involving the antiderivative of a functionLet   f  :      R    →      R  ,   f  (  x  )  =      e                  x        2            . Now, consider F an antiderivative of f.Calculate:       lim          x      →      ∞                  x      F      (      x      )              f      (      x      )      .I couldn&#039;t find the antiderivative of f, and besides finding F, I have no other ideas.

Samuel Vang · Accepted Answer

Step 1You actually need to find F. And f cannot be integrated into elementary functions anyway.Can we agree that the limit seems to be going to       ∞    ∞  ?Use L&#039;Hopitals rule.      d          d      x        x  F  (  x  )  =  F  (  x  )  +  x      F    ′    (  x  )  =  F  (  x  )  +  x  f  (  x  )        d          d      x        f  (  x  )  =  2  x  f  (  x  )Step 2After differentiating top and bottom, we have             F      (      x      )      +      x      f      (      x      )              2      x      f      (      x      )      .Still indeterminate.When we apply L&#039;Hopital&#039;s one more time, every term will have an f(x) in it, and we can cancel top and bottom.

prensistath · Accepted Answer

Explanation:By the fundamental theorem of calculus we have                     F        (        x        )        =                  ∫          0          x                          e                                    t              2                                               d        t        +        C             is an anti-derivative of f(x). Observe       lim          x      →      ∞        F  (  x  )  =  ∞.

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