INSTRUCTORS Raffi Hovasapian John Zhu

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### Solving Limits with Algebra

• Simplify until we can plug-in x without reaching an undefined value
• Method 1: Divide all terms by highest degree polynomial term
• Method 2: “unfoil”, then cancel terms in numerator and denominator

### Solving Limits with Algebra

Find limx → ∞ x
• As x approaches infinity, limx → ∞ x goes to infinity.
limx → ∞ x = ∞
Find limx → ∞ [1/x]
• Dividing a finite number by an infinitely large number will always result in zero.
limx → ∞ [1/x] = 0
Find limx → ∞ (x + [1/x])
• limx → ∞ (x + [1/x]) = limx → ∞ x + limx → ∞ [1/x] = ∞+ 0 = ∞
Find limx → ∞ [(3x2)/(x2)]
• Divide by the largest exponent
• limx → ∞ [(3x2)/(x2)] = limx → ∞ [([(3x2)/(x2)])/([(x2)/(x2)])] = limx → ∞ [3/1] = 3
3
Find limx → ∞ [3x/(x + 5)]
• limx → ∞ [3x/(x + 5)] = limx → ∞ [([3x/x])/([(x + 5)/x])]
= limx → ∞ [3/(1 + [5/x])]
= [3/(1 + 0)] = 3
3
Find limx → ∞ [(x + 5)/3x]
• limx → ∞ [(x + 5)/3x] = limx → ∞ [([(x + 5)/x])/([3x/x])]
= limx → ∞ [(1 + [5/x])/3]
= [(1 + 0)/3]
= [1/3]
[1/3]
Find limx → ∞ [(x − 4)/(x2 + 2)]
• limx → ∞ [(x − 4)/(x2 + 2)] = limx → ∞ [([(x − 4)/(x2)])/([(x2 + 2)/(x2)])]
= limx → ∞ [([1/x] − [4/(x2)])/(1 + [2/(x2)])]
= [(0 − 0)/(1 + 0)]
= [0/1]
= 0
0
Find limx → ∞ [(x2 + 500)/(x2 − 25)]
• limx → ∞ [(x2 + 500)/(x2 − 25)] = limx → ∞ [([(x2 + 500)/(x2)])/([(x2 − 25)/(x2)])]
= limx → ∞ [(1 + [500/(x2)])/(1 − [25/(x2)])]
= [(1 + 0)/(1 − 0)] = 1
1
Find limx → ∞ [(2x3 − 3x2 + 5x + 7)/(11x3 + 13)]
• limx → ∞ [(2x3 − 3x2 + 5x + 7)/(11x3 + 13)] = limx → ∞[([(2x3 − 3x2 + 5x + 7)/(x3)])/([(11x3 + 13)/(x3)])]
= limx → ∞ [(2 − [3/x] + [5/(x2)] + [7/(x3)])/(11 + [13/(x3)])]
= [(2 − 0 + 0 + 0)/(11 + 0)]
= [2/11]
[2/11]
Find limx → ∞ [(12x4 + 5x)/(x3)]
• limx → ∞ [(12x4 + 5x)/(x3)] = limx → ∞ [([(12x4 + 5x)/(x4)])/([(x3)/(x4)])]
= limx → ∞ [(12 + [5/(x3)])/([1/x])]
= limx → ∞ (12x + [5/(x2)])
= ∞+ 0 = ∞

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.