State carefully the intermediate value theorem prove that

Question

MATH 1A MIDTERM 1-

Problem 1-

(i) State the Squeeze Theorem.

(ii) Prove the Squeeze Theorem.

(iii) Using a carefully justified application of the Squeeze Theorem, find lim_x→0x⁷ sin(1/x).

Problem 2- Evaluate:

lim_x→0(√(1 + x) - √(1 - x)/x).

You should show your reasoning carefully, however you may use any of the limit laws without explanation or proof.

Problem 3-

(i) Let f be a real valued function, and let a and L be real numbers. What does it mean to say that lim_x→af(x) = L?

(ii) Prove carefully, using the definition you gave in part (i), that lim_x→1(x³ + x² + x - 5) = -2.

Problem 4-

(i) State carefully the Intermediate Value Theorem.

(ii) Prove that there is a root of the equation: 2x³ = 3^x in the interval (1, 2).

Problem 5- The figure below shows the graph of y = f(x) when

For each of the following statements, indicate if it is true or false.

(i) lim_x→4 f(x) = 3

(ii) lim_x→5^+ f(x) = ∞

(iii) lim_x→5 f(x) = ∞

(iv) lim_x→-3 f(x) exists

(v) lim_{x→- π/2^+} f(x) = ∞

(vi) lim_x→∞ f(x) = ∞

(vii) The graph y = f(x) has a horizontal asymptote at y = 0.

(viii) The graph y = f(x) has two horizontal asymptotes.

(ix) The graph y = f(x) has two vertical asymptotes.

(x) f(x) is continuous at x = 0.

(xi) f(x) is continuous at x = 1.

(xii) f(x) is continuous on the interval [1, 2].

Problem 6- Using the limit definition of derivative, show that if f(x) = x², then f'(x) = 2x.

Problem 7- What is log₄8?