MIT grad shows how to find the limit at a finite value with a square root, (sin x)/x, or absolute value. To skip ahead: 5) for a SQUARE ROOT in the numerator or denominator (to RATIONALIZE by multiplying by the "CONJUGATE"), skip to time 1:14
. 6) for a limit with something of the form (SIN X)/X, skip to time 5:38
. 7) for an ABSOLUTE VALUE in the limit expression, skip to time 14:45
5) For a SQUARE ROOT in the numerator or denominator: If you try plugging in the value for x and get a 0 in the denominator, and you cannot factor, get a common denominator, or expand to simplify the expression, then if there's a square root in the numerator or denominator, you can try MULTIPLYING by the CONJUGATE. For instance, if you have sqrt(x+1) - 3 in the numerator, you would multiply both the numerator and denominator by sqrt(x+1) + 3 because the "conjugate" just means a two-term expression with the sign flipped in front of the second term. This is a trick or technique that helps simplify because when you multiply out, or FOIL, the numerator you will get terms that cancel. It is best to leave the denominator factored, rather than multiplying out the terms since a factor is likely to cancel. Once you simplify by multiplying on top, combining like terms, and canceling any factors from the top and bottom, try plugging in the value again for x to get an actual limit value.
6) For the form (SIN X)/X in a limit expression: If you try plugging in the value that x is approaches, and you get 0 in the denominator, if your limit expression is something of the form (sin x) over x, there is a trig property that you can use to simplify. The property is that the limit of (sin x)/x, as x approaches 0, is equal to 1. If your expression isn't exactly (sin x)/x but instead has something like 2x or 3x inside the sin function, like sin(2x) over (4x), you can use the same property but first have to rearrange the expression in a way that matches what you need, as shown in the video. NOTE: Be careful not to confuse this trig property with another, very similar, (sin x)/x expression for when x is approaching infinity. That property states that the limit of (sin x)/x, as x approaches infinity, is equal to 0. Check out the video on limits at infinity for an explanation of how to use that expression.
7) For an ABSOLUTE VALUE in your limit expression: If you try plugging in the value for x and get 0 in the denominator, and you have an absolute value in your limit expression, you will probably need to re-write the limit expression using the piecewise definition of the absolute value function. You will then have an expression for the left-side limit and one for the right-side limit. If you evaluate the left side and right side, and the numbers agree, then that is your limit value. If the two sides do not have limit values that agree, then the limit does not exist.