Question

How do you locate the discontinuities of the function $y = \ln \left( {\tan \left( x \right)2} \right)$?

Answer 1

If a function is not continuous at a point in its domain, it implies that the discontinuity is present in the function and a function is continuous if the point is defined and it exists within the given range and here in same way to locate the discontinuities of the given function, find out the point of its domain, then find its discontinuity.

Complete step by step solution:
As the given function is
$y = \ln \left( {\tan \left( x \right)2} \right)$
Let us assume the given function as:
$y = \ln \left( {2\tan x} \right)$
As we know that$\tan x$ has a vertical asymptote at $x = \pm \dfrac{{n\pi }}{2}$and is zero every integer $\pi$. However, again, you could say it is continuous within its own domain because it has double-sided limits but does not ever truly reach a well-defined boundary. $\ln x$ is continuous within its own domain.
The first thing to look at is where the ln function has discontinuities. The natural log function looks at the value in parentheses and evaluates the power to which e, Euler's number, must be raised to be equal to the value in the parentheses.
For example: $\ln \left( 1 \right) = 0$ This implies that ${e^0} = 1$ which makes sense. You can also see this equivalence by making both sides exponents of $e$.
From this example, it becomes clear that we will have a problem if we take the natural log of 0. What power could $e$ be raised to make it 0, hence the answer is a discontinuity, in this case the discontinuity is negative infinity. Since there is a discontinuity in the ln function at $\ln \left( 0 \right)$, finding any points where the interior function equals 0 will be finding discontinuities. To find the rest, find where the interior function has discontinuities.

Therefore, there is a discontinuity if
$x = \pm \dfrac{{n\pi }}{2}$, $n \in Z$

Note: A function f(x) is said to be discontinuous at a point ‘a’ of its domain D if it is not continuous there. The point ‘a’ is then called a point of discontinuity of the function. $\ln x$ is continuous within its own domain (this is a tricky definition, but all it means is that its own domain happens to extend towards 0 without getting to 0, making it "continuous" because it never gets to a well-defined).