MATH 130 Fundamentals of Modern Mathematics 1 Assignment 1 2026 | University of Otago

MATH 130 Assignment 1

You may submit the assignment as often as you like (up to the due date), but only the last version will be accepted for marking. Those who correctly submit their assignment as a pdf will receive a bonus 10 marks (up to a maximum of 80 for the whole assignment).

For most of the questions a significant proportion of the marks will be awarded for presentation. Explain your working and arguments clearly, and lay out your solution so that it is easy to follow. A mathematically correct solution will not get full marks if it is poorly presented. Use the solutions in the lecture notes as a guide.

1. (20 marks) Write the following sets in interval notation.

(a) {x ∈ R : |x| ≠ 2}

(b) {x ∈ R : x² + 2x − 8 ≠ 0}

(c) {x ∈ R : |x − 2| ≥ 4}

(d) {x ∈ R : x ≥ 0} ∪ {x ∈ R : x² − 1 < 0}

2. (10 marks) Find the domains of the following functions, in interval notation.

(a) f(x) = (x² − 4x + 24) / √(x² + 5x + 6)

(b) g(x) = 2ln(x − 1) + √(4 − x)

3. (10 marks) Compute the following limits

(a) f(x) = lim (x→3) (x² − 3x) / (x² − 9)

(b) g(x) = lim (x→1) (eˣ − 7) / ln(x + 2)

4. (10 marks) Consider the function

f(x) = { 4 − ½x     if x < 2

√(x + c)     if x ≥ 2 }

(a) Find the value of c for which lim(x→2⁻) f(x) exists. (Show all working.)

(b) Sketch the graph of f. DESMOS *

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