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consider the table with function values for f of X is equal to 3 sine of 5 X over sine 2 X at X values near 0 from the table what does the limit as X approaches 0 of 3 sine of 5 X over sine of 2 X appear to be so what they have is they're approaching 0 from below here so there's negative zero point 1 and f of X is 7 point 2 3 9 5 5 0 then we get even a little bit closer to 0 X is negative 0.01 once again we're approaching 0 from values less than 0 and now we've got up to 7 point 4 9 7 3 7 5 we get even a little bit closer 0 and we get to 7 point 4 9 9 9 7 4 now let's see what happens as we approach 0 from above from values greater than 0 when x is zero point 1 at seven point two three nine five five zero actually the same as this value right over here when is zero point zero one it's seven point four nine seven three seven five actually the same values here and once again when we're at 0.001 so we're approaching from above from values greater than 0 7 point 4 9 9 9 7 4 so this is an interesting thing the limit in both cases seems to be approaching getting closer and closer to 7 point 5 and so you could imagine that this would be well let's actually graph that just to verify it for ourselves so we could type in our we could type in our answer here and check our answer and we got it right but just to visualize what that looks like let's actually get our graphing calculator so I have the graph just to show you how I get here so if I were to if I were to start on the if I were to start on the home screen just press graph define my function of X and we could just write 3 3 sine of 5 X 3 sine of 5x divided by divided by sine of 2 X and then let me define my range so see X I want it to be yes or I could make it around 0 and let's see I'm getting y-values that go up to let's see around 7 or 8 so I could make my why let me do a little bit of negative y-values so let me start at negative one for my Y range and go all the way up to let's say eight because this doesn't look like it's getting above eight or at least in the range that we're talking about and now let's graph this thing so let's see what it is okay so the graph is doing all sorts of neat things but then as we get closer and closer to zero as X gets closer and closer to 0 from the negative direction we see that the function is approaching 7.5 likewise when X is getting closer and closer to zero from the positive direction the function seems to be approaching 7.5 so you have this this this well the function isn't defined at 7.5 but it seems like or it's pretty clear that when you approach from both sides you're getting there