Math Challenges

Submissions for Problem #13

Prove the statement using the definition of a limit (epsilon and delta)

\[ \lim_{x\to2}\frac12x+3=4 \]

zapwai

Solution: \( \displaylines{\text{Given }\epsilon>0,\text{ let }\delta=2\epsilon.\\ \text{Since}\\ \left|f\left(x\right)-L\right|=\left|\frac12x+3-4\right|=\frac12\left|x-2\right|\\ \text{we have }\\ \left|x-2\right|<2\epsilon\Rightarrow\left|f\left(x\right)-L\right|=\frac12\left|x-2\right|<\frac122\epsilon=\epsilon} \)

Explanation:
We can make f(x) arbitrarily close to 4 by making x sufficiently close to 2.

fish-face

Solution: \( \)

Explanation:
|1/2(2+d) + 3 - 4| = |1 + d/2 + 3 - 4| = |d/2| For any e > 0, set d = e, then if |x - 2| < d, |1/2(x+d) + 3 - 4| = |d/2| = e/2 < e.