Question 1

Without using L’Hopital’s Rule, determine the following limits.

Question 2

(a) Determine the following right and left limits:

(b) Determine the limit

Question 3

(a) Define the function 𝑓: ℝ ⟶ ℝ by

Discuss the continuity of the function 𝑓(𝑥) at 𝑥 = 2. Explain your answer.

(b) Define the function 𝑔: ℝ ⟶ ℝ by

Discuss the continuity of the function 𝑔(𝑥). For which values of 𝑐 is the function 𝑔(𝑥) continuous? Explain your answer

(c) Let 𝑓 be a differentiable function such that 𝑓(−𝑥) = 𝑓(𝑥) for all 𝑥 ∈ ℝ.
Given that 𝑓 ′(2022) = 1, find the value of 𝑓′(−2022).

Question 4

(a) Show that the equation 𝑒 𝑥 =4 𝑥 has at least one real solution.

(b) Let 𝑓 be a differentiable function. Define a new function 𝑔 by

𝑔(𝑥) = 𝑓(𝑥) + 𝑓(3 − 𝑥). Show that 𝑔′ (𝑥) = 0 has at least one real solution.

Question 5

(a) Find the Taylor series expansion (up to the first 4 non-zero terms) of the function 𝑓(𝑥) = √𝑥 + 8 about the point 𝑎 = 0.

Use the series to estimate √8.01 3 , up to 5 significant figures.

(b) Find the point(s) on the graph of 𝑦 = 𝑥 2 + 𝑥 closest to the point (2,0). Explain your answer.