# Math-53-2nd-Long-Exam-Coverage.pdf - MATH 53 2nd Long Exam...

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MATH 53 Prepared by: DPSM Math Faculty 2 nd Long Exam Coverage
LIMITS Prepared by: DPSM Math Faculty
Definition: If f(x) gets closer and closer to a number L as x gets closer and closer to a from either side, then L is the limit of f(x) as x approaches a . The behavior is expressed by writing lim ?→𝑎 ? ? = ? Prepared by: DPSM Math Faculty
Example Evaluate Choose values of x approaching 1 and observe the values of ? ? = ? + 2 This shows that Prepared by: DPSM Math Faculty lim ?→1 ? + 2 lim ?→1 ? + 2 = 3 x 0.8 0.9 0.95 0.99 0.999 1 1.001 1.01 1.05 1.1 f ( x ) 2.8 2.9 2.95 2.99 2.999 3.001 3.01 3.05 3.10
Geometric Interpretation Prepared by: DPSM Math Faculty ? ? = ? + 2
Example Evaluate Choose values of x approaching 1 and observe the values of ? ? = ? 2 +?−2 ?−1 This shows that Prepared by: DPSM Math Faculty lim ?→1 ? 2 + ? − 2 ? − 1 lim ?→1 ? 2 + ? − 2 ? − 1 = 3 x 0.8 0.9 0.95 0.99 0.999 1 1.001 1.01 1.05 1.1 f ( x ) 2.8 2.9 2.95 2.99 2.999 3.001 3.01 3.05 3.10
Geometric Interpretation Prepared by: DPSM Math Faculty ? ? = ? 2 + ? − 2 ? − 1 Note: Limits describe the behavior of a function near a particular point, not necessarily at the point itself.
? − ? Definition Let f be a function that is defined at every number in some open interval containing a , except possibly at the number a itself. The limit of f ( x ) as x approaches a L , written as lim ?→𝑎 ? ? = ? if the following statement is true: Given any ? > 0 , however small, a number δ > 0 such that if 0 < ? − ? < ?, then ? ? − ? < ? is . Prepared by: DPSM Math Faculty
Remarks 1. The definition is saying that the function values f(x) approaches a limit L as x approaches a number a if the distance between f(x) and L can be made as small as we please by taking x sufficiently near a but not equal to a . 2. If the limit of the function exists, it is unique. 3. The term “ x approaches a ” means that x takes all values close to a but not necessarily a . 4. If lim ?→𝑎 ? ? = ? , then it means values of f(x) approaches L and L is not necessarily equal to f(a) . Prepared by: DPSM Math Faculty
Examples: Prove the following. Prepared by: DPSM Math Faculty 1. lim ?→4 ? = 4 2. lim ?→−1 3? − 1 = −4 3. lim ?→2 ? 2 + ? − 2 = 4
Exercises: Prove the following. Prepared by: DPSM Math Faculty 3. lim ?→4 ? 2 + ? − 11 = 9 1. lim ?→2 5? − 4 = 6 2. lim ?→2 ? 2 = 4
Rules on Limits For any constants a and c , and any positive integer n : i) ii) iii) Prepared by: DPSM Math Faculty lim ?→𝑎 ?? + ? = ?? + ? lim ?→𝑎 ? = ? lim ?→𝑎 ? = ?
Rules on Limits For any constants a and c , and any positive integer n : iv) If and both exist, then a) b) Prepared by: DPSM Math Faculty lim ?→𝑎 ? ? = ? lim ?→𝑎 ? ? = ? lim ?→𝑎 ? ? + ? ? = lim ?→𝑎 ? ? + lim ?→𝑎 ? ? = ? + ? lim ?→𝑎 ? ? − ? ? = lim ?→𝑎 ? ? − lim ?→𝑎 ? ? = ? − ?
Rules on Limits For any constants a and c , and any positive integer n : iv) If and both exist, then a) b) Prepared by: DPSM Math Faculty lim ?→𝑎 ? ? = ? lim ?→𝑎 ? ? = ? lim ?→𝑎 ? ? ∙ ? ? = lim ?→𝑎 ? ? ∙ lim ?→𝑎 ? ? = ? ∙ ? lim ?→𝑎 ? ? ? ? = lim 𝑥→𝑎 ? ? lim 𝑥→𝑎 ? ? = ? ? , ? ≠ 0
Rules on Limits For any constants a and c , and any positive integer n : v) vi) vii) Prepared by: DPSM Math Faculty lim ?→𝑎 ? ? ? = lim ?→𝑎 ? ? ? = ? ? lim ?→𝑎 𝑛 ?(?) = 𝑛 lim ?→𝑎 ? ? = 𝑛 ? 𝐼? lim ?→𝑎 ? ? = ? 1 ??? lim ?→𝑎 ? ? = ? 2 , ?ℎ?? ? 1 = ? 2 .
Examples: Find the following limits Prepared by: DPSM Math Faculty 1. lim ?→0 (−3? + 8) 2. lim ?→−2 5? + 7 4 3. lim ?→1 2? − 3 5 + 6?