 In this video, we'll provide the solution to question number 12 to practice exam 2 for math 1210 We're asked to prove that the limit is x approaches 0 of x squared times sine of e over x equals 0 And this is gonna be a proof we're gonna have to use a major theorem We have to identify what that theorem is give you a hint on this one on number 12 You'll be using the squeeze theorem to help you finish this thing So to use the squeeze there and we have to find an appropriate squeeze That is we have to find some type of inequality to play around with So notice if I take sine of theta for any angle whatsoever It doesn't matter which one you use the right hand side It's always gonna be bounded above by one and the left hand side is always gonna be bounded below by negative one If we replace theta instead with specifically e over x We're gonna get that sine of e over x is for the same reasons as above Bounded above by one and bounded below by negative one Then if we take this inequality here and times everything by x squared We're gonna get x squared times sine of e over x This is going to be less than or equal to one times x squared Which is x squared and it'll be greater than or equal to negative one times x squared, which is negative x squared it's important to note here that Since x squared is non-negative if you times it inequality by x squared it doesn't change the direction whatsoever So with those consideration minds consider the following if we take the limit as x approaches zero of x squared That's equal to zero That shows us that the right hand side here is gonna go off towards zero Likewise if we take the limit as x approaches zero of negative x squared That's likewise gonna equal zero that's just coming from basic continuity the limit of x squared We can just plug in zero and so maybe we should sell a little bit more steps there We're gonna negative zero squared, which we equal negative zero, which is equal to zero and likewise if we've done that to the first one We could we should have shown a little bit more detail. We're gonna get zero square, which is equal to zero so in particular the right side went to zero and the left side's also going to go to zero and so then in conclusion we can say the following here that By the squeeze theorem The squeeze theorem see Megan spell the squeeze theorem by the squeeze theorem We then conclude that the limit as x approaches zero of x squared sine of E over x is gonna equal zero and so for full credit on a question like this What are the important ingredients? So first of all you definitely need to mention that you're using the squeeze theorem because the instructions tell us Second you need to come up with the inequality the squeeze theorem requires an inequality requires that our function here is Sandwiched between two other functions if you do not provide the inequality then you can't get full credit on this type of question Then you have to show that the limits of the left side and the right side are equal to some common value And so then the squeeze there implies in that situation since the left the left side and the right side have a common limit Then that squeezes the inner function to have that same as well And that's those are the ingredients to a typical squeeze theorem argument