Can anyone me determine the infinite limit of these two problems
lim x--> -2^+(from the right)

lim x--> 0 (from both left and right)

lim x--> 2π⁻
x csc x
* i used direct substitution to determine the infinite limit of each but i have problems justifying how i got that answer by picking random points to plug into the equation

The limit is infinite because the denominator approaches 0 while the numerator does not, since when . Which infinity it approaches (positive or negative) depends on the sign of the other terms for values of near -2.

Since from the right, we're considering values of . For example, if , then ; if , then , and so on. We can keep picking values of that get closer and closer to -2, and we would see that contributes a negative sign every time. So the limit must be .

By similar reasoning above, we see that contributes a negative sign regardless of which side we approach 0 from. is always negative and is always positive, so the net effect is a negative sign and the limit from either side is .

Direct substitution gives 0 in the denominator. For we have , and for we have . Meanwhile, the numerator stays positive, which means the limit is positive or negative infinity depending on the direction in which approaches , so this limit does not exist.

what is the original number? you forgot to add it.