Limits by factoring | Limits and continuity | AP Calculus AB | Khan Academy

TL;DR

The limit of the function f(x) as x approaches 2 is 5.

Transcript

Let's say that f of x is equal to x squared plus x minus 6 over x minus 2. And we're curious about what the limit of f of x, as x approaches 2, is equal to. Now the first attempt that you might want to do right when you see something like this, is just see what happens what is f of 2. Now this won't always be the limit, even if it's defined, but it... Read More

Key Insights

• 😥 Evaluating the function at a specific point does not always give the limit.
• 👻 Simplifying the function allows for a clearer understanding of its behavior.
• 📈 Graphing the function helps visualize its characteristics and approach to a limit.
• ⛔ Numerical evaluation can also confirm the limit of the function.

Questions & Answers

Q: Why is the function f(x) undefined at x=2?

The function is undefined at x=2 because when we evaluate f(2), both the numerator and denominator become zero. This results in an undefined value.

Q: How can the function f(x) be rewritten?

The function can be rewritten as f(x) = x+3 for all values of x except x=2. This simplification allows us to better understand the graph and behavior of the function.

Q: How does the graph of f(x) look like?

The graph of f(x) is a straight line with a slope of 1 and a y-intercept of 3. It is defined for all x-values except for x=2, where it is undefined.

Q: What is the limit of f(x) as x approaches 2?

The limit of f(x) as x approaches 2 is 5. This can be confirmed both graphically, by observing the approaching values of f(x) from both sides, and numerically, by evaluating f(x) as x gets arbitrarily close to 2.

Summary & Key Takeaways

• The function f(x) is initially undefined at x=2, but it can be rewritten as x+3 for all other values of x.

• Graphically, as x approaches 2 from lower and greater values, the function seems to approach a value of 5.

• Numerically, as x gets closer to 2 from both sides, the function also approaches the value of 5.