Math 1a quiz 1 state the squeeze theorem prove the squeeze

Question

MATH 1A QUIZ 1-

Problem 1-

(i) State the Squeeze Theorem.

(ii) Prove the Squeeze Theorem.

(iii) Use the Squeeze Theorem to find

lim_x→0(x⁴/10)cos(2π/5x). Justify your answer carefully.

Problem 2-

(i) State the definition of limit for sequences (i.e. what exactly does lim_n→∞ f(n) = L mean?).

(ii) Prove that lim_n→∞ (3/4)ⁿ = 0.

(iii) Prove that lim_n→∞(n³ - 1/n³) = 1.

Problem 3-

(i) State the definition of limit for functions (i.e. what exactly does lim_x→a f(x) = L mean?).

(ii) Let f(x) = √(x - 3). Find a real number δ such that the following is true: if x is a real number such that 0 < |x - 7| < δ, then |f(x) - 2| < 1/3.

(iii) Prove that lim_x→0x⁴³ = 0.

(iv) Prove that lim_x→3x² - 4x = -3.

Problem 4- Evaluate the following limits and justify each step by indicating the appropriate Limit Laws.

(i) lim _x→-2(t² - 2/2t² - 3t + 2)³

(ii) lim_x→2√(2x² + 1/3x - 2)

Problem 5-

(i) What exactly does it mean for a function f(x) to be continuous at the point x = a?

(ii) State the Intermediate Value Theorem.

(iii) Use it to show that the polynomial p(x) = x² - πx + 2 has a root between 0 and 1.