1. Use the definition of limit to prove lim = 2x – 3 = 7.. 2. Suppose X C R, a € X', ƒ,g : X → R, lim f(x) = L, and lim g(x) = M. Use the definition of limit of a function to prove lim f(x) + g(x) = L+ M. エ→a 3. Suppose X C R, a € X', ƒ : X → R, lim f(x) = L, and c eR is a constant. Use the theorem from the “Limits of Functions" notes to prove lim cf(x) = cL. 4. Definition of Infinite Limit: Let X C R, f : X → R and a e X'. If for every M > 0 there exists & > 0 such that |f(x)| > M whenever r E X and 0 < |r – a| < 6 then we say that the limit as x approaches a of f(x) is ∞ which is denoted as lim f(x) = 0. Suppose a € R, e > 0, and f, g : N*(a, e) → R. If lim f (x) = L > 0 and lim g(x) = o, prove lim(fg)(x) = 00 エ→a

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I need help with #2 and #3. Thank you.

1. Use the definition of limit to prove lim = 2x – 3 = 7..
2. Suppose X C R, a e X', ƒ,g : X → R, lim f(x) = L, and lim g(x) = M. Use the
definition of limit of a function to prove lim f(x) + g(x) = L+ M.
エ→a
3. Suppose X C R, a e X', ƒ : X →R, lim f(x) = L, and c € R is a constant. Use the
theorem from the "Limits of Functions" notes to prove lim cf(x) = cL.
4. Definition of Infinite Limit: Let X CR, f : X → R and a e X'. If for every M > 0
there exists o > 0 such that |f(x)| > M whenever r E X and 0 < |x – a| < ô then we
say that the limit as r approaches a of f(x) is o which is denoted as lim f(x) = oo.
Suppose a e R, e > 0, and f, g : N*(a, e) → R. If lim f (x) = L > 0 and lim g(x) = 0,
prove lim(fg)(x) = 0
%3D
Transcribed Image Text:1. Use the definition of limit to prove lim = 2x – 3 = 7.. 2. Suppose X C R, a e X', ƒ,g : X → R, lim f(x) = L, and lim g(x) = M. Use the definition of limit of a function to prove lim f(x) + g(x) = L+ M. エ→a 3. Suppose X C R, a e X', ƒ : X →R, lim f(x) = L, and c € R is a constant. Use the theorem from the "Limits of Functions" notes to prove lim cf(x) = cL. 4. Definition of Infinite Limit: Let X CR, f : X → R and a e X'. If for every M > 0 there exists o > 0 such that |f(x)| > M whenever r E X and 0 < |x – a| < ô then we say that the limit as r approaches a of f(x) is o which is denoted as lim f(x) = oo. Suppose a e R, e > 0, and f, g : N*(a, e) → R. If lim f (x) = L > 0 and lim g(x) = 0, prove lim(fg)(x) = 0 %3D
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