Can use FToC to evaluate lim x → ∞ ∫ 0 x f ( t ) d t x 2 ?

Question

Can use FToC to evaluate       lim          x      →      ∞                          ∫        0        x                  f              (        t        )            d      t              x      2      ?

Jayce Bates · Accepted Answer

Because   f  (  x  )  ≥      1    2        x    2    −  2,       lim          x      →      ∞        f  (  x  )  =  ∞. Hence the limit is of the indeterminate form       ∞    ∞   and we apply L&#039;Hopital&#039;s rule. We find the derivative of the numerator by the FTC, getting      lim          x      →      ∞                          ∫        0        x            f      (      t      )      d      t              x      2        =      lim          x      →      ∞                  f      (      x      )              2      x        =      lim          x      →      ∞                  cos      ⁡      x              2      x        −      1          2      x        +      x    4  These three limits evaluate to   0 (by the squeeze theorem),   0, and   +  ∞, respectively; hence their sum has limit   +  ∞.

Can use FToC to evaluate <munder> <mo movablelimits="true" form="prefix">lim <mrow cla

Answered question

Answer & Explanation

New Questions in Integral Calculus