# Tutor profile: Jeremy S.

## Questions

### Subject: Pre-Calculus

Evaluate the limit of $$3x^5+9x^2+8$$ as $$x$$ approaches $$-\infty$$.

While this problem looks confusing, there are some interesting math tricks we can do to get around this! When evaluating limits, we substitute these limit values into our variables. For infinity, this gets tricky. As x approaches infinity, the values we substitute get larger and larger. But, if we had a variable in a denominator, and no variables in the numerator, then the number would get smaller! So in order to get a set number, let's factor out our largest variable term, $$x^5$$. If we factor one of of these out of every term, we are left with $$3+\frac{9}{x^3}+\frac{9}{x^5}$$. When you evaluate this limit, your second and third term would become closer to 0 as x approaches infinity, so we can say these numbers become 0. This means that the value of the limit is 3.

### Subject: Geometry

You go with your family to visit a historical lighthouse. The lighthouse stands 209 feet tall. Unfortunately, you can only get 63 feet from the lighthouse. From the distance you're standing, what angle are you looking up at the bulb from?

First, a great strategy to employ when solving any problem is to draw a picture. There are two objects we're concerned with: the lighthouse bulb and you. You're standing 63 feet from the lighthouse, so draw a line representing that linear distance. However, the bulb is at the top of the lighthouse, which is 209 feet off the ground, so at one end of that line, draw a line perpendicular to that line representing the height of the lighthouse. Now we don't need to know how far you are from just the bulb, but we need to know the angle you're making with the ground, so draw a line connecting the top of the lighthouse with where you are, creating a triangle. But you should see that this triangle is a special kind of triangle -- it's a right triangle! This means that we can use our trig functions to find the missing angle. We can use trig functions to find angle measures in right triangles if we already have at least two side lengths. But which two side lengths? Let's think about the sides in relation to the angle we're looking for. For one, we don't have the hypotenuse of the triangle. But, we do have the closest side length, the "adjacent" side, and the furthest side length, the "opposite" side. Since we have both of these, we use tangent (think SOH CAH TOA -- I have "opposite" and "adjacent" so I use tangent; T. O. A.). Since we're looking for the actual angle, we have to use inverse tangent, $$tan^{-1}$$. The final step is to calculate $$tan^{-1}(\frac{209}{63})$$, which is equal to $$73.23^{\circ}$$. This means that you're looking up at the lighthouse at a 73 degree angle, which is pretty steep -- don't hurt your neck!

### Subject: Algebra

Solve the following equation for $$x$$: $$2x+4 =x-6$$

To solve for $$x$$, start by subtracting one $$x$$ term from each side. This is done to "isolate" the variable, $$x$$, or getting it by itself one on side of the equation. It can be either side, but it has to be by itself so that we know what it is equal to. Also, we have to keep both sides of our equation equal, otherwise we're changing the value of an equation and we can't solve it. Finally, in equations, we have constant terms and variable terms, and they can't combine by addition or subtraction, so subtracting an $$x$$ from each side means that you're subtracting $$x$$ from each of the variable terms on that side. So on the right side, you subtract $$x$$ from $$x$$, but not from $$-6$$, leaving you with just $$-6$$ on that side. On the left side, you subtract $$x$$ from $$2x$$, leaving the left side as $$x+4$$ and the current expression as $$x+4=-6$$. Now, to solve, there is just one more step. Subtract $$4$$ from each side, following the same rules as above. Now, notice that, on the left side of the equation, $$x$$ is isolated and on the right side, $$-4$$ and $$-6$$ can combine to $$-10$$, meaning that $$x=-10$$.