MHB Finding Continuous Values of $\frac{e^{sinx}}{4 - \sqrt{x^2 - 9}}$

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The function $\frac{e^{\sin x}}{4 - \sqrt{x^2 - 9}}$ is continuous where the denominator does not equal zero and the expression under the square root is non-negative. The conditions for continuity lead to the inequalities $x^2 - 9 \geq 0$, resulting in $x \in (-\infty, -3] \cup [3, \infty)$, and $4 - \sqrt{x^2 - 9} \neq 0$, which gives $|x| \neq 5$. Thus, the function is continuous for $x \in (-\infty, -5) \cup (-5, -3] \cup [3, 5) \cup (5, \infty)$. The initial reasoning was incorrect due to miscalculating the conditions and not considering the non-negativity of the radicand. The correct intervals for continuity have been established.
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I have this function

$$\frac{e^{sinx}}{4 - \sqrt{x^2 - 9}}$$

And I need to find all the values for which this function is continuous.

So I do

$$4 - \sqrt{x^2 - 9} \ne 0$$

$$ \sqrt{x^2 - 9} \ne 4 $$

$$ x^2 - 9 \ne 16 $$

$$ x^2 \ne 7 $$

And therefore, the function is not valid where

$$ x \ne +/- \sqrt{7} $$

However, this appears to be completely wrong. Why is my reasoning wrong?
 
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First, I would look at:

$$x^2-9\ge0$$

This gives us:

$$(-\infty,-3]\,\cup\,[3,\infty)$$

Next, look at

$$\sqrt{x^2-9}\ne4$$

$$x^2\ne25$$

$$|x|\ne5$$

$$(-\infty,-5)\,\cup\,(-5,5)\,\cup\,(5,\infty)$$

And so, we find:

$$(-\infty,-5)\,\cup\,(-5,-3]\,\cup\,[3,5)\,\cup\,(5,\infty)$$
 
MarkFL said:
First, I would look at:

$$x^2-9\ge0$$

This gives us:

$$(-\infty,-3]\,\cup\,[3,\infty)$$

Next, look at

$$\sqrt{x^2-9}\ne4$$

$$x^2\ne25$$

$$|x|\ne5$$

$$(-\infty,-5)\,\cup\,(-5,5)\,\cup\,(5,\infty)$$

And so, we find:

$$(-\infty,-5)\,\cup\,(-5,-3]\,\cup\,[3,5)\,\cup\,(5,\infty)$$

So what I had was correct but not comprehensive?
 
tmt said:
So what I had was correct but not comprehensive?

No, you took:

$$x^2-9\ne16$$

And added 9 to the left side while subtracting 9 from the right side. You also did not require the radicand to be non-negative. :)
 
tmt said:
I have this function

$$\frac{e^{sinx}}{4 - \sqrt{x^2 - 9}}$$

And I need to find all the values for which this function is continuous.

So I do

$$4 - \sqrt{x^2 - 9} \ne 0$$

$$ \sqrt{x^2 - 9} \ne 4 $$

$$ x^2 - 9 \ne 16 $$

$$ x^2 \ne 7 $$
No, if $x^2- 9\ne 16$ then $x^2= x^2- 9+ 9= 16+ 9= 25$, not 16- 9= 7.

And therefore, the function is not valid where

$$ x \ne +/- \sqrt{7} $$

However, this appears to be completely wrong. Why is my reasoning wrong?
 
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