Lesson Find the values of a parameter such that the given function is continuous at x=0

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<H2>Find the values of a parameter such that the given function is continuous at x=0</H2>

<H3>Problem 1</H3>Find the value(s) of a parameter 'k' such that the following piecewise-defined function is continuous at x=0:

<pre>
    f(x) = {{{x/(k + sin(e^(1/x)))}}}, if x =/= 0 ,

f(x) =  0, if x=0.
</pre>

<B>Solution</B>

<pre>
Look at the denominator in the function definition. It is  k + sin(e^(1/x)).

Let's assume   -1 <= k <= 1.

If x is positive and tends to zero, x --> 0, then  1/x  tends to infinity.

Hence, the expression  e^(1/x)  tends to infinity, too, and is a monotonically increasing function.

It implies that  sin(e^(1/x))  takes the value of -k infinitely many times.

So, the denominator  k + sin(e^(1/x)) takes the value of zero infinitely many times, as x becomes smaller 
and approaches 0.

It tells us that in the interval  -1 <= k <= 1, the given function has infinitely many poles
as x tends to zero.

In other words, if k is between -1 and 1, the given function is not continuous at x = 0.

Vice versa, if k is less than -1 or k is greater than 1, the denominator of the given function 
is a continuous, bounded function, which NEVER takes the value of zero.

This means that for all values of k outside the interval [-1,1], the given function is continuous at zero.

<U>ANSWER</U>.  The given function is a continuous function at x = 0 for all values of k outside the interval [-1,1].
</pre>

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