4. If an + 0, prove that |an+1| lim inf |an+1| < lim inf Jan| < lim sup lan] <_lim sup |anl |anl n→+∞ n→+∞ n→+∞ n→+∞

Q: 1. Use the definition to prove that lim_ (r + 8)² = 25

A:

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A: Concept:

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Q: 4) Find each limit a) lim(x3 – 5x - 1) b) lim *_2-4 (5p) x-2 x-2 x-2 c) lim *2-,3x-9 d) lim v2x2 + 3…

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Q: Evaluate and state limit laws used

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A: We have to check

Q: Asap

A: Let's evaluate limits.

Q: anx" + ... + ajx + ao_ with a. >0 and bk > 0, show that brak + 5. If f(x) = lim f(x) = +∞ + b1x + bo…

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Q: う lim -3K -×)。 X 00

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A: Given: limx→1x-1x-1

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A: Given,One has to prove the above expression using  definition.

Q: 3* –1 lim V1+x -1,

A:

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A: Given- limx→1x2-1x10-1

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A:

Q: 15. a) Find the limit. x²-9 lim x--3x²+2x-3 b)lim √25+h-5 h

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A:

Q: Consider lim exp 8118 7-x³ x+∞ x² +7x a) lim b) If we let t t lim = 3 2 = → 7-x³ x² +7x 7-x³ x² + 7x…

A: To find:-

Q: 8.) lim (-1+ x² – x³ + 3x*) 9.) x3-2x-1 lim x-3 co-X 2x2+3x+1 lim X+ 3x-5x+2 10.)

A:

Q: Find the limits

A: Step 1: Given, find the limit for: limx→2​(x2−4x2−x−2​)Let's factorize both numerator and…