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The concept of limits allows us to study the behavior of the function as x gets closer and closer to a given point in this case 3 , even though we cannot evaluate it at exactly that point. This is denoted in limit notation as: There are a number of different ways to evaluate the limit of a function at a given point, including graphically, numerically, or in some cases, by simply evaluating the limit at the given point.

It also provides the means for us to discuss another far-reaching concept in calculus, that of continuity. Continuity Functions can be either continuous or discontinuous. Informally, a function is said to be continuous if its graph is a single unbroken curve with no holes. Referencing the figure above, at the point 3, 6 , one would have to lift their pencil to draw the graph, so it is discontinuous.

On the other hand, the figure below is an example of a continuous function: It is possible to trace the entire function without the need to raise the pencil. Estimate derivatives numerically and apply rules to compute derivatives of a variety of algebraic functions, including higher-order derivatives. Understand the relationship between differentiability and continuity. Use implicit differentiation to solve related rates problems. Approximate values of functions using linearization. Relate the graph of a function to properties of its derivative.

Solve optimization problems by using the derivative to find extrema of functions. Integration Evaluate Riemann sums and interpret them geometrically and contextually. Compute the integral of a function as the limit of a Riemann sum.

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Understand the relationship between differentiability and continuity. Use implicit differentiation to solve related rates problems. Approximate values of functions using linearization. Relate the graph of a function to properties of its derivative.

Solve optimization problems by using the derivative to find extrema of functions. Integration Evaluate Riemann sums and interpret them geometrically and contextually. Compute the integral of a function as the limit of a Riemann sum. Relate integrals and antiderivatives through the fundamental theorem of calculus. In this page I'll introduce briefly the ideas behind these concepts. These ideas are explored more deeply in the links below. The Idea of Limits of Functions When we talked about functions before, we payed attention at the values of functions at specific points.

The idea behind limits is to analyze what the function is "approaching" when x "approaches" a specific value. To start getting used to this idea, let's turn to this graph: When x approaches the value "a" in the x axis, the function f x approaches "L" in the y axis.

In this graph I drawed a big pink hole at the point a,L. Let's focus on the point 1,1. We can see from the graph that when x approaches 1, the function f x approaches 1. When this happens, we say that: This is read "the limit as x approaches 1 of x squared equals 1". Why Limits are Useful You might ask what this is useful for. Very good question.

Why would you need to know what the function is approaching? You already know the function equals 1 when x equals 1, right? As an example of this, let's consider the following function: Don't let this notation intimidate you! This only means that this function equals x2 when x is anything other than 1, and equals 0 when x equals 1.

Let's see what the graph looks like: What does the function approach when x approaches 1?