Can you prove that limxrarr1 x 1 using the epsilon delta

Question

Midterm 1 Review Worksheet - Math 1A, section 103

1. What is lim_x→1 x?

2. Can you prove that lim_x→1 x = 1 using the ε, δ definition of a limit?

3. Find the domain of the function √x + log(3 - x).

4. Find a formula for the inverse of the function f(x) = 2^{3^x}. What is the domain of f^-1?

5. Evaluate the limit

lim_x→5 x- - 25/x - 5.

6. Evaluate the limit

lim_x→5 2^x/x - 5.

7. Show that the equation cos(x) = x has at least one positive real solution.

8. Show that lim^{x→0^+} log₁₀(2x) = -∞ using the definition of a limit.

9. We know that lim_x→10 log₁₀(x) = 1. Given ε > 0 with ε < 1, find δ so that whenever |x - 10| < δ, we can guarantee that |log₁₀(x) - 1| < ε.

10. Valentine's Day Bonus: Is there a function whose graph looks like a heart? Why or why not? Can you find an explicit equation whose graph looks like a heart? (For instance, the graph of x² + y² = 1 is a circle, which is getting close.)