Determine if the given function can be extended to a continuous function at x = 0. If so, approximate the extended
function's value at x = 0 (rounded to four decimal places if necessary). If not, determine whether the function can be
continuously extended from the left or from the right and provide the values of the extended functions at x = 0.
Otherwise write "no continuous extension."
-$$f ( x ) = \frac { \tan x } { x }$$
A) f(0) = 1 only from the right
B) No continuous extension
C) f(0) = 1
D) f(0) = 1 only from the left

Question

Quizplus · Accepted Answer

The function $\frac{\tan x}{x}$ can be continuously extended at $x = 0$ by using the limit process. As $x$ approaches 0, both the numerator ($\tan x$) and the denominator ($x$) approach 0. Using L'Hôpital's Rule, we differentiate the numerator and the denominator separately, which gives us $\frac{\sec^2 x}{1}$. Evaluating this at $x = 0$ gives $\sec^2(0) = 1$, indicating that the function can be continuously extended at $x = 0$ with a value of 1.

Determine If the Given Function Can Be Extended to a Continuous