Understanding limits is a key idea in calculus. It helps us see how functions act, especially when they get close to certain points or even infinity.
So, what are limits when it comes to functions?
What are Limits?
A limit is a value that a function gets close to as you change the input (or variable) to a specific number. For example, if we want to find the limit of a function ( f(x) ) as ( x ) gets close to 2, we write it like this: ( \lim_{x \to 2} f(x) ).
Continuous Functions:
Many functions are continuous, which means you can draw them without lifting your pencil. For a function to be continuous at a certain point, the limit as you get close to that point must match the value of the function at that point. Take the function ( f(x) = 3x + 1 ) as an example. When we find the limit as ( x ) approaches 1, we see that ( \lim_{x \to 1} f(x) = 4 ). This is the same as saying ( f(1) = 4 ).
Special Cases:
Sometimes, functions have holes or jumps in them. For instance, the function ( g(x) = \frac{x^2 - 1}{x - 1} ) can be rewritten as ( g(x) = x + 1 ), but it doesn't work when ( x = 1 ). At that point, the function is undefined. However, if we look at the limit as ( x ) approaches 1, we find that it is 2, even though ( g(1) ) isn’t a real number!
By learning about limits, students in Year 9 start to understand calculus better. They get to see how different functions behave in a deeper way.
Understanding limits is a key idea in calculus. It helps us see how functions act, especially when they get close to certain points or even infinity.
So, what are limits when it comes to functions?
What are Limits?
A limit is a value that a function gets close to as you change the input (or variable) to a specific number. For example, if we want to find the limit of a function ( f(x) ) as ( x ) gets close to 2, we write it like this: ( \lim_{x \to 2} f(x) ).
Continuous Functions:
Many functions are continuous, which means you can draw them without lifting your pencil. For a function to be continuous at a certain point, the limit as you get close to that point must match the value of the function at that point. Take the function ( f(x) = 3x + 1 ) as an example. When we find the limit as ( x ) approaches 1, we see that ( \lim_{x \to 1} f(x) = 4 ). This is the same as saying ( f(1) = 4 ).
Special Cases:
Sometimes, functions have holes or jumps in them. For instance, the function ( g(x) = \frac{x^2 - 1}{x - 1} ) can be rewritten as ( g(x) = x + 1 ), but it doesn't work when ( x = 1 ). At that point, the function is undefined. However, if we look at the limit as ( x ) approaches 1, we find that it is 2, even though ( g(1) ) isn’t a real number!
By learning about limits, students in Year 9 start to understand calculus better. They get to see how different functions behave in a deeper way.