I need to prove the limit using the definition of limit lim x → c f ( x ) = L ↔ ∀ ϵ &gt; 0 ∃ δ &gt; 0 : 0 &lt; | x − c | &lt; δ → | f ( x ) − L | &lt; ϵ

Question

I need to prove the limit using the definition of limit      lim          x      →      c        f  (  x  )  =  L  ↔  ∀  ϵ  &amp;gt;  0    ∃  δ  &amp;gt;  0  :  0  &amp;lt;      |    x  −  c      |    &amp;lt;  δ  →      |    f  (  x  )  −  L      |    &amp;lt;  ϵ

Arcatuert3u · Accepted Answer

Since   tan  ⁡  (  x  ) has an asymptote at   x  =      π    2  , we should stipulate at the outset a maximum value of delta which is less than the distance from the point we are working at and the asymptote (this distance is       π    4  ), for instance   δ  &amp;lt;      π    8  . So in what follows we will consider only   x  ∈  (      π    4    −      π    8    ,      π    4    +      π    8    )  =  (      π    8    ,            3      π        8    ). Now, if we suppose that   ϵ is arbitrary and   0  &amp;lt;      |    x  −      π    4        |    &amp;lt;  δ holds, we have      |    tan  ⁡  (  x  )  −  1      |    =      |                  sin        ⁡        (        x        )                    cos        ⁡        (        x        )              −                  sin        ⁡                  (                      π            4                    )                            cos        ⁡                  (                      π            4                    )                      |    =      |                  sin        ⁡        (        x        )        cos        ⁡                  (                      π            4                    )                −        sin        ⁡                  (                      π            4                    )                cos        ⁡        (        x        )                    cos        ⁡        (        x        )        cos        ⁡                  (                      π            4                    )                      |    =                    |            sin      ⁡              (        x        −                  π          4                )                    |                            |            cos      ⁡      (      x      )              |            cos      ⁡              (                  π          4                )            Now in the numerator we can use the inequality       |    sin  ⁡      (    x    −          π      4        )        |    ≤      |    x  −      π    4        |   while to bound the denominator we notice that cosine is decreasing on the interval   (      π    8    ,            3      π        8    ) and so       1                  |            cos      ⁡      (      x      )              |              &amp;lt;      1          cos      ⁡      (                        3          π                8            )      . These two inequalities allow us to write                    |            sin      ⁡              (        x        −                  π          4                )                    |                            |            cos      ⁡      (      x      )              |            cos      ⁡              (                  π          4                )              &amp;lt;                    |            x      −              π        4                    |                    cos      ⁡              (                              3            π                    8                )            cos      ⁡              (                  π          4                )              &amp;lt;      δ          cos      ⁡              (                              3            π                    8                )            cos      ⁡              (                  π          4                )              ≤  ϵIf we chose   δ  =  min  {      π    8    ,  cos  ⁡      (                  3        π            8        )    cos  ⁡      (          π      4        )    ϵ  }

I need to prove the limit using the definition of limit <munder> <mo movablelimits="true" fo

Answered question

Answer & Explanation

New Questions in Calculus 1