Question

knowledge & understanding (8 marks)
true/false
question 1 (1 point)
a limit exists if both the right - hand and left - hand limits are equal.
true
false
question 2 (1 point)
an infinite discontinuity means a function is merely a set of scattered points.
true
false

Explanation:

Question 1

Step1: Recall limit existence condition

By the definition of the existence of a limit in calculus, for a function \(y = f(x)\) at a point \(x = a\), \(\lim_{x
ightarrow a}f(x)\) exists if and only if \(\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)\). That is, the left - hand limit \(\lim_{x
ightarrow a^{-}}f(x)\) (as \(x\) approaches \(a\) from values less than \(a\)) and the right - hand limit \(\lim_{x
ightarrow a^{+}}f(x)\) (as \(x\) approaches \(a\) from values greater than \(a\)) are equal.

Question 2

Step1: Define infinite discontinuity

An infinite discontinuity of a function \(y = f(x)\) at \(x = a\) occurs when either \(\lim_{x
ightarrow a^{-}}f(x)=\pm\infty\) or \(\lim_{x
ightarrow a^{+}}f(x)=\pm\infty\). A function with a set of scattered points is a discrete function (for example, a function defined only for integer values of \(x\) like \(y = n\) where \(n\in\mathbb{Z}\)). An infinite discontinuity is a type of non - removable discontinuity related to the behavior of the function near a point (the function values tend to infinity), not a set of scattered points.

Answer:

Question 1: True
Question 2: False