Ashesi University
Course Name: Engineering Calculus
Activities & Problem Solving
Module 2 – Limits
Date: Jan. 19, 2026
Learning Objectives:
• Using correct notation to describe the limit of a function.
• How to find the limit of functions using graphical, tabular, and algebraic methods.
• Explain the relationship between one-sided and two-sided limits.
• Apply the idea of limits to Asymptotes and Continuity.
Motivation:
•The idea of a limit underlies the various branches of calculus. It is, therefore,
appropriate to begin our study of calculus by investigating limits and their properties.
•The vision of human exploration by NASA to distant parts of the universe illustrates
the idea of space travel at high speeds. But, is there a limit to how fast a spacecraft
can go?
•Pupil adjustment of your eyes when you enter a very dark or bright room, like how a
camera behaves.
Trial 1.1: How do you solve it algebraically? Hint: Simplify when direct substitution is
not possible.
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Trial 1.2: Use tabular, graphical, and algebraic evidence to find the value for each
limit.
Trial 1.3: Evaluate the indicated limit, if it exists. Assume that 𝐥𝐢𝐦
𝒙→𝟎
𝒔𝒊𝒏(𝒙)
=1
𝒙
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Limits at infinity (Vertical, Horizontal and Oblique Asymptotes)
The line 𝑦 = 𝐿 is called a horizontal asymptote of the curve 𝑦 = 𝑓(𝑥) is either
The line x= 𝑎 is called a vertical asymptote of the curve 𝑦 = 𝑓(𝑥) if
Some curves have oblique asymptotes, that is, neither horizontal nor vertical. Where
𝑚 ≠ 0, then the line 𝑦 = 𝑚𝑥 + 𝑏 is called a slant asymptote if
Examples: The following two examples illustrate the use of horizontal, vertical, and
slant asymptote.
(a)
(b)
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Trial 1.4:
1
1. Find any horizontal asymptotes to the graph of 𝑓(𝑥) = 2 − 𝑥 .
The line 𝑦 = 2 is a horizontal asymptote.
2. Determine all vertical and horizontal
3.
4. asymptotes of the following:
5.
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6.
7.
8.
5|Page