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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

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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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I need help with #2 and #3. Thank you.

1. Use the definition of limit to prove \(\lim_{{x \to 5}} 2x - 3 = 7\). 2. Suppose \(X \subseteq \mathbb{R}, a \in X'\), \(f, g : X \to \mathbb{R}\), \(\lim_{{x \to a}} f(x) = L\), and \(\lim_{{x \to a}} g(x) = M\). Use the definition of limit of a function to prove \(\lim_{{x \to a}} (f(x) + g(x)) = L + M\). 3. Suppose \(X \subseteq \mathbb{R}, a \in X'\), \(f : X \to \mathbb{R}\), \(\lim_{{x \to a}} f(x) = L\), and \(c \in \mathbb{R}\) is a constant. Use the theorem from the “Limits of Functions” notes to prove \(\lim_{{x \to a}} cf(x) = cL\). 4. **Definition of Infinite Limit**: Let \(X \subseteq \mathbb{R}, f : X \to \mathbb{R}\) and \(a \in X'\). If for every \(M > 0\) there exists \(\delta > 0\) such that \(|f(x)| > M\) whenever \(x \in X\) and \(0 < |x - a| < \delta\) then we say that the limit as \(x\) approaches \(a\) of \(f(x)\) is \(\infty\) which is denoted as \(\lim_{{x \to a}} f(x) = \infty\). Suppose \(a \in \mathbb{R}, \epsilon > 0\), and \(f, g : N^*(a, \epsilon) \to \mathbb{R}\). If \(\lim_{{x \to a}} f(x) = L > 0\) and \(\lim_{{x \to a}} g(x) = \infty\), prove \(\lim_{{x \to a}} (fg)(x) = \infty\).
Transcribed Image Text:1. Use the definition of limit to prove \(\lim_{{x \to 5}} 2x - 3 = 7\). 2. Suppose \(X \subseteq \mathbb{R}, a \in X'\), \(f, g : X \to \mathbb{R}\), \(\lim_{{x \to a}} f(x) = L\), and \(\lim_{{x \to a}} g(x) = M\). Use the definition of limit of a function to prove \(\lim_{{x \to a}} (f(x) + g(x)) = L + M\). 3. Suppose \(X \subseteq \mathbb{R}, a \in X'\), \(f : X \to \mathbb{R}\), \(\lim_{{x \to a}} f(x) = L\), and \(c \in \mathbb{R}\) is a constant. Use the theorem from the “Limits of Functions” notes to prove \(\lim_{{x \to a}} cf(x) = cL\). 4. **Definition of Infinite Limit**: Let \(X \subseteq \mathbb{R}, f : X \to \mathbb{R}\) and \(a \in X'\). If for every \(M > 0\) there exists \(\delta > 0\) such that \(|f(x)| > M\) whenever \(x \in X\) and \(0 < |x - a| < \delta\) then we say that the limit as \(x\) approaches \(a\) of \(f(x)\) is \(\infty\) which is denoted as \(\lim_{{x \to a}} f(x) = \infty\). Suppose \(a \in \mathbb{R}, \epsilon > 0\), and \(f, g : N^*(a, \epsilon) \to \mathbb{R}\). If \(\lim_{{x \to a}} f(x) = L > 0\) and \(\lim_{{x \to a}} g(x) = \infty\), prove \(\lim_{{x \to a}} (fg)(x) = \infty\).
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