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1.9-1.10 Hole vs VA Core Rule
How do you tell if x=a creates a HOLE or a VERTICAL ASYMPTOTE?
Does the factor cancel?
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1.9-1.10 Hole Definition
A HOLE occurs at x=a when...
Think: numerator and denominator
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1.9-1.10 VA Definition
A VERTICAL ASYMPTOTE occurs at x=a when...
Think: numerator and denominator
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1.9-1.10 Limits at a Hole
Near a HOLE at x=a, what are lim(x to a-) and lim(x to a+)?
Both sides, same value
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1.9-1.10 Limits at a VA
How do you find whether lim(x to a) is +inf or -inf near a VA?
Track signs of each factor
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1.9-1.10 Even vs Odd Mult VA
Even-mult VA: both limits go ___? Odd-mult VA: limits go ___?
Same direction or opposite?
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1.9-1.10 Example 2b
g(x) = (x-2)(x+4)/[(x-2)(x-3)]. lim(x->2-) and lim(x->2+) = ?
Factor (x-2) cancels!
Plug into simplified form
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1.9-1.10 Example 3a
Build f(x) with: hole at x=3 giving limit 5, VA at x=1 going -inf/+inf.
f(x) = k(x-3)/[(x-3)(x-1)]
Solve for k
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Answer: Hole Definition
Factor in denominator CANCELS with numerator factor
Undefined at x=a, but both one-sided limits equal the same finite y-value. Graph shows an open circle at (a, y).
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Answer: Hole vs VA Core Rule
Cancel = HOLE No cancel = VERTICAL ASYMPTOTE
Step 1: Factor completely. Step 2: Check each denominator factor. Step 3: If it cancels, hole. If not, VA.
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Answer: Limits at a Hole
Both limits equal the SAME finite value
Cancel the common factor first. Then plug x=a into the simplified form. Example: (x-2)(x+4)/[(x-2)(x-3)] at x=2 gives 6/(-1) = -6.
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Answer: VA Definition
Factor in denominator does NOT cancel with numerator
Function goes to plus or minus infinity near x=a. The limits may be opposite (odd mult) or same (even mult).
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Answer: Example 2b Limits
lim left = -6 lim right = -6 (Hole at x=2)
(x-2) cancels -> hole. Simplified: (x+4)/(x-3). At x=2: 6/(-1) = -6. Both sides equal -6.
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Answer: Limits at a VA
Track sign of each factor; if result is positive -> +inf, negative -> -inf
Near VA: substitute x = a-epsilon and a+epsilon. Track +/- for each factor. pos/pos or neg/neg = positive; pos/neg = negative.
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Answer: Example 3a Answer
f(x) = 10(x-3) / [(x-3)(x-1)]
Hole at x=3: (x-3) in both. VA at x=1: (x-1) in denominator only. Simplified k/(x-1). At x=3: k/2=5 -> k=10.
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Answer: Even vs Odd Mult VA
Even: SAME direction (both +inf or both -inf) Odd: OPPOSITE directions
Even multiplicity: expression stays same sign on both sides -> both limits same. Odd: sign flips -> one +inf, one -inf.