Q. 55

Question

Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .

$\lim_{x \to 1} (2 x^{2} - 4 x + 3) = 1$ .

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Answer

Delta-epsilon proof for $\lim_{x \to 1} (2 x^{2} - 4 x + 3) = 1$ is, whenever $0 < |x - 1| < δ$ , we also have $|(2 x^{2} - 4 x + 3) - 1| < ε$ .

1Step 1 . Given information

$\lim_{x \to 1} (2 x^{2} - 4 x + 3) = 1$ .

2Step 2 . Consider ε > 0 , choose δ = ε 2 .

For all $x$ with $0 < |x - 1| < δ$ , we also have $|(2 x^{2} - 4 x + 3) - 1| < ε$ . $|(2 x^{2} - 4 x + 3) - 1| = |2 x^{2} - 4 x + 3 - 1| = |2 x^{2} - 4 x + 2| = 2 |x^{2} - 2 x + 1| = 2 |{(x - 1)}^{2}| = 2 {|x - 1|}^{2} < 2 {(δ)}^{2} = 2 {(\sqrt{\frac{ε}{2}})}^{2} = ε$

Therefore, whenever $0 < |x - 1| < δ$ ,we also have $|(2 x^{2} - 4 x + 3) - 1| < ε$ .

Q. 54

Q. 56

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