MTH107: Without using L’Hopital’s Rule, determine the following limits: Calculus I Homework, SUSS

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 ????: ℝ ⟶ ℝ by

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

(b) Define the function ????: ℝ ⟶ ℝ by

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

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

Question 4

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

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

????(????) = ????(????) + ????(3 − ????). Show that ????′ (????) = 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 ????(????) = √???? + 8 about the point ???? = 0.

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

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

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