Define a function $f$ as follows: $f ( 0 ) = 0$ and, for any $x > 0$, $$f ( x ) = \lim _ { L \rightarrow \infty } \int _ { \frac { 1 } { x } } ^ { L } \frac { 1 } { t ^ { 2 } } \cos ( t ) \, d t$$ (or, in simpler notation, the improper integral $\int _ { \frac { 1 } { x } } ^ { \infty } \frac { 1 } { t ^ { 2 } } \cos ( t ) \, d t$). (i) Show that the definition makes sense for any $x > 0$ by justifying why the limit in the definition exists, i.e., why the improper integral converges. (ii) Find $f ^ { \prime } \left( \frac { 1 } { \pi } \right)$ if it exists. Clearly indicate the basic result(s) you are using. (iii) Using the hint or otherwise, find $\lim _ { h \rightarrow 0 ^ { + } } \frac { f ( h ) - f ( 0 ) } { h }$, i.e., the right hand derivative of $f$ at $x = 0$. We can take the limit only from the right hand side because $f ( x )$ is undefined for negative values of $x$. Hint: Break $f ( h )$ into two terms by using a standard technique with an appropriate choice. Then separately analyze the resulting two terms in the derivative.
Define a function $f$ as follows: $f ( 0 ) = 0$ and, for any $x > 0$,
$$f ( x ) = \lim _ { L \rightarrow \infty } \int _ { \frac { 1 } { x } } ^ { L } \frac { 1 } { t ^ { 2 } } \cos ( t ) \, d t$$
(or, in simpler notation, the improper integral $\int _ { \frac { 1 } { x } } ^ { \infty } \frac { 1 } { t ^ { 2 } } \cos ( t ) \, d t$).
(i) Show that the definition makes sense for any $x > 0$ by justifying why the limit in the definition exists, i.e., why the improper integral converges.
(ii) Find $f ^ { \prime } \left( \frac { 1 } { \pi } \right)$ if it exists. Clearly indicate the basic result(s) you are using.
(iii) Using the hint or otherwise, find $\lim _ { h \rightarrow 0 ^ { + } } \frac { f ( h ) - f ( 0 ) } { h }$, i.e., the right hand derivative of $f$ at $x = 0$. We can take the limit only from the right hand side because $f ( x )$ is undefined for negative values of $x$.\\
Hint: Break $f ( h )$ into two terms by using a standard technique with an appropriate choice. Then separately analyze the resulting two terms in the derivative.