To find:The value of lim_{x \rightarrow 0} (\frac{4^{3x}-4^x}{4^x-1}) if it exists. If limit does not exist ,then find out if one-sided limit exists

How to find the axes of a rotated ellipse
I have a set of 17 points which I know are on an ellipse. I have the x,y co-ordinates of each point; the y-axis is vertical and the x-axis is horizontal. I want to prove these points are on an ellipse, but the ellipse is rotated clockwise by approximately 14 degrees (determined visually - I want to calculate the exact value of the rotation). I need to find the exact position of the major and minor axes (x' and y') and I do not know the values of the semi-major axis (a) or the semi-minor axis (b). Is this possible?
I have tried to find the x,y co-ordinates of the points furthest from (and nearest to) the centre of the ellipse, but I've only managed this through very many tedious iterations and it isn't exact -- is there a better way? Thank you.

The type of conic sections for the nondegenerate equations given below.

a) ${\displaystyle 8x^2-2y^2-3x+2y-6=0}$

b) ${\displaystyle -6y^2+4x-12y-24=0}$

c) ${\displaystyle -9x^2-4y^2-18x+12y=0}$

To fill: The missing form of the expression

Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work. ${\displaystyle r=\frac{10}{5+2\cos \theta }}$

The current in a wire varies with time according to the relation ${\displaystyle I=55A?\left(0.65\frac{A}{s^2}\right)t^2}$.
How many coulombs of charge pass a cross section of the wire in the time interval between ${\displaystyle t=0}$ and ${\displaystyle t=8.5s}$ ?
Express your answer using two significant figures.
What constant current would transport the same charge in the same time interval?
Express your answer using two significant figures.

How do you name the curve given by the conic ${\displaystyle r=6}$?

Fill the blank in the given statement.