How do you take the limit of a composite piecewise function f(f(x))?
Only the graph is given.
lim f(f(x))
x->2
I figured the way to do it was to find the lim f(x) = C then find lim f(x).
x->2 x-> C
The simple questions on Khan academy's "Limits of composite functions" on khan academy corroborate with my method in the case of two different functions. But this does not seem to work, according to the given answer.
have you seen this? https://www.youtube.com/watch?v=fjcHK6GGqm8
To find the limit of a composite piecewise function f(f(x)), you need to approach it step by step. Here's the process:
Step 1: Evaluate the inner function f(x) as x approaches 2. Look at the graph to determine the behavior of f(x) as x approaches 2 from both the left and the right. If f(x) is continuous at x = 2, then f(2) is simply the value of the function at x = 2. If f(x) is not continuous at x = 2, you'll need to consider the one-sided limits.
Step 2: Once you have the value of f(2), substitute it back into the original function to evaluate f(f(x)).
Step 3: Evaluate the outer function f(f(x)) as x approaches 2. Again, observe the behavior of f(f(x)) as x approaches 2 from both the left and the right. If f(f(x)) is continuous at x = 2, the limit will simply be the value of the function at x = 2. If f(f(x)) is not continuous at x = 2, consider the one-sided limits.
Remember to consult the graph of the function to understand its behavior and determine the value of f(2) properly. If you need help with the specific function, please provide additional information or the actual graph so that I can assist you further.
To take the limit of a composite piecewise function f(f(x)), you need to evaluate the function at the desired limit point and follow the function's definition based on the given graph.
Here's an approach to take the limit of a composite piecewise function f(f(x)) using the given graph:
1. Start by identifying the relevant intervals on the graph of f(f(x)) around the limit point x = 2. Look for the intervals where x is approaching 2 from the left and from the right.
2. Determine the value of f(f(x)) in each of these intervals. To do this, use the graph to find the corresponding y-values for each x-value in the intervals.
3. Evaluate the limit of f(f(x)) as x approaches 2 by considering the limits from both the left and the right sides. Check if the functions have the same limit value as x approaches 2 from both directions.
4. If the limit values from both sides match, then the limit of f(f(x)) as x approaches 2 exists and is equal to that limit value. If the limit values differ, the limit does not exist.
It's important to note that the approach you mentioned, finding the limit of f(x) and then taking the limit of f(x) as x approaches C, may not always yield the correct answer. In the case of composite functions and piecewise functions, it is essential to consider the graph and evaluate the limit directly based on the given conditions.
If you have the specific function or graph you would like to discuss, please provide that information, and I can help you further.