Lesson Notes By Weeks and Term v4 - SHS 1

Lesson Video

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Performance objectives

• Define left-hand and right-hand limits.
• Calculate one-sided limits using algebra.
• Understand limits using graphs.
• Know the condition for a limit to exist.

Lesson summary

This lesson introduces the foundational concept of Calculus: the limit. We will explore the idea of "approaching" a value without necessarily reaching it. Think about the speedometer in a car; it shows your speed at a single instant. How can you have speed in an instant with no time passing? This is a question limits help us answer. In Ghana, we can think about economic trends, like the price of cocoa approaching a certain value before a major announcement, or the rate at which water fills the Akosombo dam as it gets closer to its maximum level. Understanding limits is the first step to understanding change, which is what Calculus is all about.

Lesson notes

2.1 The Idea of "Approaching" a Value

Imagine you are walking along a straight path towards a friend standing at the 5-metre mark. You can approach your friend from two directions: From the left: You might be at the 4m mark, then 4.5m, 4.9m, 4.99m, getting closer and closer to 5 from numbers *less than* 5. From the right: You could be at the 6m mark, then 5.5m, 5.1m, 5.01m, getting closer and closer to 5 from numbers *greater than* 5.

In Calculus, the concept of a limit is about the value a function *approaches* as the input (usually `x`) gets closer and closer to a certain number from either side. 2.2 Left-Hand and Right-Hand Limits

When we look at the behaviour of a function `f(x)` as `x` approaches a number `a`, we need to consider the approach from both sides. The Left-Hand Limit: This is the value that `f(x)` gets closer to as `x` approaches `a` from the left side (i.e., through values less than `a`). Notation: We write this as `lim_{x \to a^-} f(x) = L`. The small minus sign (`-`) in the superscript of `a` means "from the left". The Right-Hand Limit: This is the value that `f(x)` gets closer to as `x` approaches `a` from the right side (i.e., through values greater than `a`). Notation: We write this as `lim_{x \to a^+} f(x) = R`. The small plus sign (`+`) in the superscript of `a` means "from the right". 2.3 Graphical Interpretation of One-Sided Limits

Evaluation guide

• Classify left -hand and right- hand limits algebraically and with the aid of technology where
• available or through any creative means.
• Group discussion:
• meaning of approaching a number on a real number line from the left or from the right of the number.