A bag contains 6 red marbles, 6 white marbles, and 4 blue marbles. Find P(red or blue).

1. perpendicular

2. parallel

3. perpendicular

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right now, attempting to use the quotient limit law is not possible since both the individual limits of the numerator and denominator are 0.

note that we can rationalize the numerator by multiplying the numerator and the denominator by . our motivation is to get rid of the roots so that we can try to cancel (x-4) since that is what causes the 0/0 situation.

since

this cancellation is only valid for x ≠ 4, but it is okay here since the limit only takes into account values around x = 4, not at x = 4.

now we make the following observation:

if x is close to 4 but more than 4, we have that

this ultimately means that

so overall, we have 8/(small positive number), meaning that the entire fraction

is a large positive number. intuitively we get our answer:

(note: this does not mean that the limit exists, nor are we seeing ∞ as an actual number. this above is just shorthand for saying that the values of the expression grow arbitrarily large as x gets closer to a, but more than a.)