Using sagemath"s jupyter notebook language code solve the following:

Sagemath is capable of working with piecewise defined functions,

a$)$ Find which function in Sage is used to create piecewise functions and define the piecewise function

$f(x)=\{\begin{array}{ccc}{x}^2& if& x<mn>1</mn>\\ 6& if& x=1\\ 5-3x& if& x>1\end{array}$

b$)$ Try to compute $\underset{x1}{lim}f(x),$ and observe that we get an error. However, the function can use multiple algorithms. We can change the algorithm

used by adding an additional argument algorithm $=\text{'}$sympy$\text{'}$ or algorithm $=$ 'giac'. Try to compute $\underset{x1}{lim}f(x)$ using each of these algorithms.

Notice that while there is no longer an error, the value returned is incorrect $($what should the output be$?).$ Does anything change when we try to compute

the limit on the left or right with these algorithms?

Remark: While we see that limit$()$ does not work correctly for piecewise functions, there are ways to circumvent this issue efficiently. By using the

function expression$\_$at$()($the syntax is f$.$expression$\_$at$()),$ we can take the limit of the appropriate piece of $f.$ For example, the input

$.$ expression$\_$at$(2),$ x$=1)$ will return the correct limit from above for $f.$