Can you evaluate this limit?
Yikes! The answer is nice, but finding it is a bear! L'Hopital will work, of course, but you need to apply it more than half a dozen times. (Dozens of chain rules without making a mistake...) You might try numerical approximation, but both the numerator and denominator get really small really fast, so it is hard to calculate accurate numbers when "x" is close to zero (every x-value much smaller than one hundredth incorrectly gives "undefined"). A spreadsheet approach is not very convincing. Sketching a graph is fairly convincing. Just don't zoom in TOO much or you start to see weird things (resulting from round off errors). My best suggestion for exact results is to use Maclaurin Series. Unfortunately, you need to use at least FOUR terms: Sin(x) = x - x^3 / 6 + x^5 /120 - x^7 / 5040 + O(x^9) Tan(x) = x + x^3 / 3 + 2x^5 / 15 + 17x^7 / 315 + O(x^9) ArcSin(x) = x + x^3 / 6 + 3x^5 /40 + 5x^7 / 112 + O(x^9) ArcTan(x) = x - x^3 / 3 + x^5 / 5 - x^7 / 7 + O(x^9) Substitute to find Sin(Tan(x)) and Tan(Sin(x)) and ArcSin(ArcTan(x)) and ArcTan(ArcSin(x)). If you can do that (the x^7 term is the killer and it IS needed), then it is easy to subtract and finally divide to get a linear approximation for: [sin(tan(x)) - tan(sin(x))] / [arcsin(arctan(x)) - arctan(arcsin(x))] THEN the limit is trivial, just substitute x=0.
How can I avoid silly mistakes in exams keeping in mind the time limit?
Silly mistakes are the worst thing than can happen to students specially in competitive exams.Avoiding silly mistakes is not an easy task.you have to practice more questions for it because pepole use to learn fast with their own mistakes.Read question carefuly focus on the word like correct/incorrect/not incorrect/not correct etc.In Physics and Chemistry double checks the units.Stay active while solving the questions keep away yourself from distractions like paper level,your performance in exam and any previous experience.OMR filling pattern is most important.according to me you should fill your OMR simultaneously because in this method there is minimum chances of mass mistake and you don't have to think about the extra time for OMR filling.Time limit is very important. In exam there are some questions which are specially formatted for wasting the time of students.always solve those kind of questions in last because they may kill time for 3–5 easy questions.Remember one thing Every mark matters ..
Can you evaluate the limit for (cosx) /x?
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How can I avoid silly mistakes in Mathematics?
Main Reasons for careless Mistakes is Habit. We have only 5 or 10 days of exams in a year. Our Mind does not has habit of exam. So an untrained mind/unprepared mind can take random decisions for immediate situations occur at that time and it is natural we can commit mistakes in exams. Other Important Reasons are lack of Practice, lack of Concentration, exam Fear and Pressure, improper Sleeps during exams, Mental Fatigue, Time limitation, Try to solve Questions very speedily. To avoid silly mistakes these steps may help you:One thing you should know 80-90 % PRESSURE WE FEEL is DUE TO TIME LIMIT and NOT DUE TO THE EXAM HALL. (You can check this fact. Give exam at home with time limit. You will feel the pressure. )So you should:a) Give enough mock tests before final exams. If you have time give at least 50 mock test in coaching or at home with time limit and discipline. During these mock test you will do many mistakes, analyse them and they will not be repeated in Final Exam.b) Always work hard at the starting of preparation, slowly slowly you may decrease this.And in the month of final exam study very reasonable according to your capacity. Relax and have confidence.If you did your preparation in very systematic and planned way then in the month of exam you will be naturally very confident. So no stress, no pressure, no mental fatigue. Your mind will be fresh, your concentration will be very high, and your presence of mind, your logic, your common sense all will give you 100% positive result.c) Give 10-15 mock tests at the exact time of your final exam (means if your exam time is 10-1 pm give mock test during this time daily.What will it do. It will activate your Biological Clock of Body and Mind. If you will apply this then on the final exam you mind, your concentration all other tools will automatically switch on exact this time.d) Make strategies for exams assuming different situations: What will you do if questions are out of course. What will you if paper is very tough?What will you do if paper is very lengthy? How will you manage the time?What will you do if you find some questions you are not prepared for that?What will you do if you have committed a mistake in exam?What will you do if you feeling lack of concentration or feeling nervousness? So assume possible situations that may occur and possible best solutions you may apply. I hope this will helpful for you.Please share your feedback if it work for you.DO YOUR BEST!!!!