For this one, I already know that csc(x) is 1/sin(x) since csc is the inverse of sin. The limit of csc(x) or 1/sin(x) as theta approaches 0 from the right hand side is infinity. Adding a constant, which in this case is 5, should still leave the answer as infinity. Thus would infinity be the correct answer or is there more to it?
For this one I am told to find 3 different limits for that function.
a) As x approaches -10 from the left.
b) As x approaches -10 from the right.
c) As x approaches -10 from both sides.
For (a), -10 is being approached from the left, which means that we are looking at values smaller than -10, but close to it. For example, -10.1 is a candidate. When I plug in -10.1 into the formula, I get -2231.1. When I put in -10.01, I get an even larger negative number. -10.001 yields an even LARGER negative number, and so on. Thus would it be safe to say that as x approaches -10 from the left, the limit is negative infinity?
For (b), -10 is being approached from the right, which means that we are looking at values that are LARGER than -10, but close to it. I used -9.9 for the first candidate. When I plug it into the function, I get 2169.1. When I plug in -9.99 I get 21969.01. The larger I make it, as long as its larger than -10, the larger the answer it yields. Thus would it be correct to say that as x approaches -10 from the right, the limit is positive infinity?
For (c), based on what I wrote above, the limit should not exist since the right hand and left hand limits differ. Plus, trying to plug in -10 itself into the function would result in the denominator being 0, thus making it undefined.
Is my reasoning correct? If not please nudge me in the right direction. Thanks in advance to whoever helps out.