Indeterminate Forms and L’Hospital’s Rule

 As x approaches a certain number from both sides – what does y approach?  In order for them limit to exist you must approach the same y from both sides of x

i. Direct Substitution: ii. Factor Cancel: iii. Rationalize:

i. Squeeze Theorem ii. Trig Substitution

 Direct substitution gives which we call the indeterminate form.  To fix an indeterminate we divide by the highest power of x in the denominator

 The limit of a function at is it’s horizontal asymptote

 Not all indeterminate forms can be evaluated by algebraic manipulation. This is particularly true when both algebraic and transcendental functions are involved. In cases like these, use L’Hospital’s Rule

 Under certain conditions the limit of the quotient is determined by the limit of  Let f and g be functions that are differentiable on an open interval (a,b) containing c, except possibly at c itself. If then provided the limit on the right exist (or is infinite).

 Direction Substitution gives so we are allowed to use L’Hospital’s Rule