Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

$\underset{x\to 9}{lim}\frac{\sqrt{x}-3}{x-9}$

ka1leE
2021-10-14
Answered

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

$\underset{x\to 9}{lim}\frac{\sqrt{x}-3}{x-9}$

You can still ask an expert for help

cyhuddwyr9

Answered 2021-10-15
Author has **90** answers

To find $\underset{x\to 9}{lim}\frac{\sqrt{x}-3}{x-9}$

We rationalize the numerator as

$\underset{x\to 9}{lim}\frac{\sqrt{x}-3}{x-9}\times \frac{\sqrt{x}+3}{\sqrt{x}+3}$

$=\underset{x\to 9}{lim}\frac{x+3\sqrt{x}-3\sqrt{x}-9}{(x-9)(\sqrt{x}+3)}$

$=\underset{x\to 9}{lim}\frac{x-9}{(x-9)(\sqrt{x}+3)}$

$=\underset{x\to 9}{lim}\frac{1}{\sqrt{x}+3}$

$=\frac{1}{\sqrt{9}+3}$

$=\frac{1}{6}$

Therefore,$\underset{x\to 9}{lim}\frac{\sqrt{x}-3}{x-9}=\frac{1}{6}$

We rationalize the numerator as

Therefore,

Jeffrey Jordon

Answered 2022-07-04
Author has **2313** answers

asked 2020-12-16

The figure shows the surface created when the cylinder ${y}^{2}+{Z}^{2}=1$ intersects the cylinder ${x}^{2}+{Z}^{2}=1$. Find the area of this surface.

The figure is something like:

asked 2022-04-06

If (2,6) lies on the curve $f\left(x\right)=a{x}^{2}+bx$ and y=x+4 is a tangent to the curve at that point. Find a and b?

asked 2022-04-14

How do you find the equation of the tangent line to the curve $f\left(x\right)={x}^{2}+2x$ ; at x=3, x=5?

asked 2022-06-24

Does the antiderivative of an indicator function?

I have an indicator function of the form $I\{a<t<b\}$ and I need the antiderivative of it with respect to t.

I believe I can't use the fundamental theorem of calculus to simply write it as ${\int}_{0}^{t}I\{a<s<b\}ds$ (which would be very convenient) since it's not a continuous function.

It's not clear to me whether such a function even exists or how I would write it.

I have an indicator function of the form $I\{a<t<b\}$ and I need the antiderivative of it with respect to t.

I believe I can't use the fundamental theorem of calculus to simply write it as ${\int}_{0}^{t}I\{a<s<b\}ds$ (which would be very convenient) since it's not a continuous function.

It's not clear to me whether such a function even exists or how I would write it.

asked 2021-02-21

Use the method of your choice to evaluate the following limits.

$\underset{(x,y)\to (1,0)}{lim}\frac{y\mathrm{ln}y}{x}$

asked 2022-01-16

Compute the first-order partial derivatives.

$z={x}^{2}+{y}^{2}$

asked 2021-06-29

Find derivatives of the functions defined as follows.

$y={e}^{-2x}$