Now that we have understood what the limit of a function is, we must now know the various properties surrounding the limits of a function.

There are 7 main properties of limits that we must know in calculus and vectors.

lim k = k, for all constants of k
x->a

lim x = a
x->a

lim [f(x) +/- g(x)] = lim f(x) +/- lim g(x)
x->a x->a x->a

lim [cf(x)] = c[lim f(x)], for all constants of c
x->a x->a

lim [f(x)g(x)] = [lim f(x)] [lim g(x)]
x->a x->a x->a

lim f(x) = lim f(x), given that lim g(x) does not equal to 0
x->a g(x) x->a x->a
lim g(x)
x->a
(The property above this statement is meant to represent what to do when two limits are being divided with each other.)

lim [f(x)]^n = [lim f(x)]^n, for any value of n that is a rational number.
x->a x->a

Due to constraints regarding Steam's implementation of textboxes (almost seems like they weren't made for mathematical calculations,) our example will unfortunately be very simple. However, I urge those of you intrigued enough to read to here to attempt problems involving these properties yourself.

Evaluate lim (3x^2+ 4x - 1)
x->2
Solution :
= lim (3x^2) + lim (4x) - lim (1)
x->2 x->2 x->2

= 3 lim (x^2) + 4 lim (x) - 1
x->2 x->2

= 3lim x ^2 +4(2) - 1
x->2

= 3(2)^2+8-1
= 19

When we are solving for limits algebraically, we can use the techniques of:
- Direct Substitution
- Factoring
- Rationalizing
- One-sided limits
- Change of variables

For these techniques, you may also use a graph and/or a table in order to check your result.

Thank you for reading our stupid reviews I hope in the future Calculus and Vectors will grow to finally become a military force and allow me to finally achieve total world domination.




Calculate the maximum amount of reviews Steam will allow us to post as a Steam curator. (Hint: Not enough)

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To understand more about calculus, we need to learn more about functions themselves, and their limits.

A limit in a function occurs when it can approach a certain x value, however it can never cross it. A good comparison would be like a parabola; getting closer to the number, but never equaling or becoming greater.

Finding the limit of a function requires two components; the function itself, and before the equation itself, 'lim' (which stands for limit), and below it, what value 'x' is approaching. For example, an equation like this would be

lim (x^2 + 2x - 3)
x->4

The limit can be technically speaking any real number, except your answer cannot be DNE, as in, anything divided by zero.

Now, to find the limit itself, we can use many ways and strategies to do so.

First is direct substitution, where we can simply substitute in the number 'x' is approaching towards into the equation. This of course, has limitations of its effectiveness, as only with simplistic functions, can we apply this strategy.

We could then try to applying what we know from past years of mathematics.

In situations where direct substitution will equal to DNE, we can use the following steps to solving the limit:

1. Expand the numerator.
   
2. If there is a quadratic in your denominator, simplify.
   
3. Factor your new equation.
   
4. Cross out anything possible.
   
5. Direct substitution to solve.

NOTE: Sometimes, your numerator has a difference of cubes or a difference of squares; watch out for those!

Your function may has a square root in it. That's okay, as we can use a method known as multiplying by the conjugate.

To do so, multiply both sides by the side that has the square root (If it has a number outside of the square root, inverse its sign.) From there, most things cross out itself on its own, and you can then apply direct substitution.

Using these methods laid out for you above, you can push functions to the limit!

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Calculus is the study of rates of change, as in, finding the slope on a curved function.

Slope, the change in y divided by the change in x.
"change in" is also known as delta (Δ).

We usually find slope by obtaining two points on the graph, and inputting it into the slope equation. This is commonly referred to as "rise (y value) over run (x value)".

Although slope equation is the standard and the most used to find the rate of change, we can also use secant and tangent lines on a graph.

A secant line is a line that touches the curve at two points. It is used to attain two certain points to find the particular average rate of change.

A tangent line is a line that "just touches" the curve at a singular point to represent instantaneous rate of change. To note though, tangent lines are known to be inaccurate due its reliability on your ability to draw an accurate line, therefore it is subject to human error.

Through this introduction towards calculus, you will be able to branch towards more complex concepts and ideas towards a full understanding in this new world of mathematics.

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