AP Calculus AB: 1.9 Connecting Multiple Representations of Limits - Exam Style questions with Answer- MCQ

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Question

Let $g$ be a function that is increasing for $x less than 1$ and increasing for $x greater than 1$ A I only

B II only

C III only

D I and III only

▶️Answer/Explanation

Ans:D

Since $\lim_{x\rightarrow 1}\frac{x^{2}+3x-4}{x-1}=\lim_{x\rightarrow 1}\frac{(x-1)(x+4)}{(x-1)}=\lim_{x\rightarrow 1}(x+4)=5$ ,the expression in (I) could represent the function $g$ .The graph in (III) could also represent the function $g$ , since the limit determined by the graph as $x$ approaches 1 is equal to 5. This is the graph of the function in (I).The table in (II) could not represent the function $g$ , since the values in the table suggest that the function grows without bound approaching $x equals 1$

x = 1

from both sides, and thus the limit would not exist at $x equals 1$

Question

If $h$ is a piecewise linear function such that $\lim_{x\rightarrow 3}h(x)$ does not exist, which of the following could represent the function $h$ ?

A I Only

B II only

C III only

D I and III only

▶️Answer/Explanation

Ans:D

The table in (II) could represent the function $h$ . The data in the table for $x less than 3$ are linear with slope 2, so the limit as $x$ approaches 3 from the left would be 7. The data in the table for $x greater than 3$ are also linear with slope $negative 1$ , so the limit as $x$ approaches 3 from the right would be 4. Since the left- and right-hand limits are not equal, the limit at $x equals 3$ would not exist for the function represented by this table.The graph in (III) could also represent the function $h$ , since the graph shows a jump discontinuity at $x equals 3$ , where the left- and right-hand limits are unequal. The values in the table in (II) are the same as the values on this graph.The expression in (I) could not represent the function $h$ , since $\lim_{x\rightarrow 3^{-}}(2x+1)=7$ and $\lim_{x\rightarrow 3^{+}}(10-x)=7$ implies that the limit of the expression in (I) exists at $x equals 3$ .