MIT grad shows how to find any limit as x approaches a finite value/constant value (and not infinity). To skip ahead: 1) For an example of PLUGGING IN/SUBSTITUTION, skip to time 1:45. 2) For FACTORING to simplify, skip to 3:53. 3) For GETTING A COMMON DENOMINATOR, skip to time 8:09. 4) For EXPANDING by opening up parentheses to simplify and find the limit, skip to 12:01. Nancy formerly of MathBFF explains the steps.
Jump to the PART 2 video (https://youtu.be/v9fQ_QeCHpI) for how to find the limit for: 5) a SQUARE ROOT in the numerator or denominator (to RATIONALIZE by multiplying by the "CONJUGATE"); 6) something of the form (SIN X)/X; or 7) an ABSOLUTE VALUE in the limit expression.
For LIMITS at INFINITY, jump to: https://youtu.be/nViVR1rImUE
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1) TRY PLUGGING IN/SUBSTITUTION: The first way to try to find the limit value is to plug in for x. In the limit expression, x is approaching a certain number. If you plug in this number and get a value that is defined, then that is your limit. HOWEVER, if you get ZERO in the denominator when you plug in, then you have not found the limit yet and need to try something else to find the limit value.
2) TRY FACTORING: If you plugged in the value for x, and you got zero in the denominator (or the form 0 over 0), check whether you can factor and simplify to find the limit. If the limit expression is made up of a polynomial in a numerator and a polynomial in the denominator, then it is a very good idea to try factoring because a factor in the top may cancel with a factor in the bottom to give you a simpler expression. Then, plugging into this simpler expression may give you an actual limit value.
3) TRY GETTING A COMMON DENOMINATOR: If you plugged in the value for x, and you got zero in the denominator, and you cannot factor the expression, you have to try something else. If your limit expression has fractions within a fraction ("a complex rational expression"), try getting a common denominator in the expression. Use algebra to get a common denominator between the two fractions that are in the numerator (or denominator), and when simplifying, terms may cancel so that you have a simpler expression you can plug into to get a limit value.
4) TRY EXPANDING/OPENING UP PARENTHESES: Again, if you plugged in and got a zero in the denominator, and you can’t factor or get a common denominator, consider opening up parentheses and expanding expressions by FOIL-ing or multiplying out and combining like terms. Simplifying in this way may lead to a simpler expression you can plug into to get a limit value.
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