Essential Knowledge 1.1A1
Essential Knowledge 1.1A1 Students will know that given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c). If the limit exists and is a real number, then the common notation is
.
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Introduction
"Given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c)."
Mathematicians have created two main definitions for a limit. Mathematicians use the definition above to describe the idea of a limit in a way that is good enough for most people.
(Here's the exact definition of a limit, known as the "epsilon-delta definition of a limit", which is what I learned in AP Calculus. I don't think it helped me understand limits very much though, so I don't lament its removal from the AP Calculus test.)
I'll color code the definition to use in my explanation.
"Given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c)."
Understanding the Symbols
Most parts of the definition have a mathematical symbol. Using common notation, aka "symbolically".
Notice that the only part that doesn't have a symbol is "...if... can be made arbitrarily close to..." That's because this is something that must be true before one can even write a limit.
Understanding the Definition
The basic idea of a limit is pretty simple. Let's use the color coded definition again for some examples.
Example 1
"Given a function f,..."
Let's use f(x) = x2 + 1 for our function. Remember this is basically the same as y = x2 + 1.
"...the limit of f(x) as x approaches c... by taking x sufficiently close to c (but not equal to c)."
Let's use c = 5. This means we'll use numbers for x that approach 5, but never quite reach 5. The following table shows this using numbers smaller than 5.
We also need to approach 5 using numbers greater than 5.
"...is a real number R if f(x) can be made arbitrarily close to R..."
Both approaches show f(x) clearly getting closer and closer to 26. How close? Arbitrarily close--in other words, as close as we want. We can get closer and closer to 26 the closer we make x to 5.
Therefore, this limit exists, and it is equal to 26. According to the definition of a limit, R = 26. We can say
"The limit of x2 + 1 as x approaches 5 equals 26."
or symbolically
The graph below also shows this
Example 2
So far limits probably seem pretty silly. Like, why not just substitute x = 5, duh? Lets do an example where the limit can be more useful than substitution.
"Given a function f,..."
Let's use f(x) = for our function.
"...the limit of f(x) as x approaches c... by taking x sufficiently close to c (but not equal to c)."
Let's use something for "c" that calculus is all about: infinity (which has the symbol ∞).
Now, if you tried to solve this just using substitution, for x = ∞ would give you , which is indeterminate and therefore kinda useless.
Using limits, this means we'll use numbers for x that just keep getting bigger and bigger.
You can't have a number greater than infinity so we can't make a second table.
"...is a real number R if f(x) can be made arbitrarily close to R..."
The table above shows f(x) clearly getting closer and closer to 0. How close? Arbitrarily close--in other words, as close as we want. We can get closer and closer to 0 the larger we make x.
!! And 0 is a much more useful answer than indeterminate.
Therefore, this limit exists, and it is equal to 0. According to the definition of a limit, R = 0. We can say
"The limit of as x approaches ∞ equals 0."
or symbolically
The graph below also shows ln(x) / x getting closer and closer to 0 as x gets larger
Example 3
Finally, let's look at a limit that doesn't exist
"Given a function f,..."
Let's use for our function.
"...the limit of f(x) as x approaches c... by taking x sufficiently close to c (but not equal to c)."
Let's use the other number that calculus is all about, 0.
Now, if you tried to solve this just using substitution, for x = 0 would give you , which is undefined (because anything divided by 0 is undefined) and therefore kinda useless.
Using limits, this means we'll use numbers for x that just keep getting closer and closer to 0.
We also need to approach 0 using numbers greater than 0.
"...is a real number R if f(x) can be made arbitrarily close to R..."
The tables above do not show f(x) clearly getting closer and closer to any number. In fact, looking at either table, values of f(x) are all over the place, getting bigger, smaller, positive, negative. This makes sense, because we know sin(x) just keeps oscillating between -1 and 1 as x increases, so it will never actually settle on any number.
Therefore, this limit does not exist. We can say
"The limit of as x approaches 0 does not exist
or symbolically
The graph below shows how, as x approaches 0, the graph just starts oscillating up and down like crazy, so the limit could never hope to settle on an actual number