Got a midterm worksheet. Realized how confused I am.

Here are some questions:

lim x^3 + 12x^2 - 5x/5x
x->0

Find the indicated limit:
lim 11x/absolute value x
x->0+

I am not sure if you mean

lim x^3 + 12x^2 - 5x/5x
x->0
or
lim (x^3 + 12x^2 - 5x) /5x
x->0
but either way
as x ---> 0 you end up with -5x/5x
which is -1
all those higher order terms go to zero faster than 5x
like if x = .1
x^2 = .01
x^3 = .001

Now if you are given that you are on the POSITIVE side of the origin, then |x| = x

then
11x/x = 11 for any old x if x>0

To find the limit of a function, you need to evaluate the function as x approaches a specific value. Let's solve each of these problems step-by-step:

1. lim(x^3 + 12x^2 - 5x)/(5x) as x approaches 0:

In this problem, we have a rational function. To evaluate it, we can simplify it first by factoring out the common term in the numerator, which is x:

lim (x * (x^2 + 12x - 5))/(5x)

Now, we can cancel out the common factor, "x":

lim (x^2 + 12x - 5)/5

At this point, if we substitute x = 0 directly into the expression, we get an indeterminate form (0/0). To resolve this, we can use algebraic manipulation.

Now, we'll factor the numerator:

lim ((x+5)(x-1))/5

Since the limit is as x approaches 0, we can substitute the value of x into the factored expression:

lim ((0+5)(0-1))/5
lim (-5)/5 = -1

Therefore, the limit of the function as x approaches 0 is -1.

2. lim(11x)/(|x|) as x approaches 0+ (from the right-hand side):

Here, we have an absolute expression in the denominator that poses a problem when x approaches 0, as the function becomes undefined. To evaluate this limit, we can use the concept of a one-sided limit, where we approach x from the right-hand side.

As x approaches 0 from the right side (x > 0), the absolute value function can be simplified as |x| = x. Therefore, the expression becomes:

lim(11x)/x

Now we can cancel out the common factor of x:

lim 11

Hence, the limit of the function as x approaches 0 from the right-hand side (x > 0) is 11.