$\lim_{x\to16}\frac{4-\sqrt{x}}{16x-x^2}$
I am not sure what to do, I have tried factoring everything and using both conjugates, neither options gives me anything usable.
$\lim_{x\to16}\frac{4-\sqrt{x}}{16x-x^2}$
I am not sure what to do, I have tried factoring everything and using both conjugates, neither options gives me anything usable.
First pull out the obvious factor of $x$ in the denominator to get $\frac{4-\sqrt x}{x(16-x)}\;.$ The $x$ in the denominator won’t cause any problems in taking the limit, so focus on the rest: $\frac{4-\sqrt x}{16-x}\;.$ Notice that each term in the denominator is the square of the corresponding term in the numerator: $\frac{4-\sqrt x}{16-x}=\frac{4-\sqrt x}{4^2-(\sqrt{x})^2}\;.$ That last denominator is the difference of two squares; what do you know about factoring such differences?
Let's try to get things to look the same: $ {4-\sqrt x\over 16x-x^2}={4(1-{\sqrt x\over4})\over 16x(1-{x\over 16} ) }= { 1-{\sqrt x\over4} \over 4x(1-{x\over 16} ) }. $ Observe that $({\sqrt x\over 4})^2={x\over 16}$ for $x>0$.
Can you see how to take advantage of the formula $a^2-b^2=(a+b)(a-b)$?
Alternative Method: (Assuming one knows differentiation)
$\lim_{x\to16}\dfrac{4-\sqrt{x}}{16x-x^2}=\lim_{x\to16}\frac1x\dfrac{4-\sqrt{x}}{16-x}=\lim_{x\to16}\frac1x\dfrac{\sqrt{16}-\sqrt{x}}{16-x}.$ By definition, $\lim\limits_{x\to16}\frac{4-\sqrt{x}}{16-x}$ is the derivative of $\sqrt{x}$ evaluated at $x=16$. And since the derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$, it follows that its value is: $\frac{1}{2\sqrt{16}}=\frac{1}{8}$. Thus $\lim\limits_{x\to16}\frac{4-\sqrt{x}}{16-x}=\tfrac18$, hence the value of the limit is $\tfrac1{16}\cdot\tfrac18=\tfrac{1}{128}$.
Try using L'Hôpital's rule in the case of an indeterminate form.
One way is by a rationalizing substitution: $ \begin{align} u & = \sqrt{x} \\ \\ u^2 & = x \\ \\ \text{As }x\to16, & u \to 4. \end{align} $ So $\lim_{x\to16}\frac{4-\sqrt{x}}{16x-x^2} = \lim_{u\to4}\frac{4-u}{16u^2-u^4} = \lim_{u\to 4} \frac{4-u}{u^2(4-u)(4+u)} = \lim_{u\to4} \frac{1}{u^2(4+u)}=\frac{1}{4^2(4+4)}=\frac{1}{128}.$