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[h] SL 5.1 Concept of a limit

[q] **Limits**

We have seen limits a couple of times in this course, in functions. We saw graphs that approached an asymptote: This is the limit of f(x). We also saw the limit of the sum of a geometric sequence, if |r| < 1. In both of these cases, we never actually reach the limit, we just get infinitely close and never go beyond it. Limits will be very important in calculus.

[q]

Slopes of Curves

What is the slope of this curve? Well, it varies. We could maybe ask what the slope is at x₁, but that is still quite complicated. Previously, we needed two points if we wanted to calculate the slope. So maybe we could choose another point [x₁ + h] and find a slope from there. It still doesn’t seem very accurate, and it looks like if we move x₁ + h closer to x, this straight line becomes more appropriate.

[a]

In fact, if we just consider the limit of this slope as h tends to 0, that line has the correct slope. [The slope/gradient function] = \(\lim_{h \to 0} \frac{f(x + h) – f(x)}{h}\)

Now, actually evaluating limits requires some more advanced math, but we will actually just learn rules for various types of functions.

As h → 0, the straight line that is used to find the slope is called a tangent: a line that only touches the curve once, at x₁.

[q]

When that formula is evaluated, the result is a function for the gradient of the tangent, which will vary depending on the x-value considered.

Notation

Given a function, its gradient function is called: the derivative.

Function: f(x) = _____

Derivative: f'(x) = _____

You can see the rise/run connection here.

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