Limit of Function f(x) = x^3 at x=1

To find the limit of the function f(x) = x^3 at x=1, we need to analyze the behavior of the function as x approaches 1.

Graphical Explanation

We can plot the graph of f(x) = x^3 and observe the trend as x approaches 1.



From the graph, we can see that as x approaches 1 from both sides, the function approaches the value of 1. Therefore, we can say that the limit of f(x) at x=1 exists and is equal to 1.

Mathematical Explanation

We can also prove the limit mathematically using the epsilon-delta definition of a limit.

Let ε be any positive number and δ be a positive number such that |x-1| < δ.="" />

Then, we have

|f(x) - 1| = |x^3 - 1|

= |(x-1)(x^2 + x + 1)|

Since |x-1| < δ,="" we="" have="" />

|x^2 + x + 1| < 3δ^2="" +="" 3δ="" +="" 1="" />

So, if we choose δ = min{1, ε/7}, then we have

|f(x) - 1| < ε="" />

Therefore, by the epsilon-delta definition of a limit, we can say that the limit of f(x) at x=1 exists and is equal to 1.

Conclusion

In conclusion, we can say that the limit of the function f(x) = x^3 at x=1 exists and is equal to 1. We can prove this both graphically and mathematically using the epsilon-delta definition of a limit.