Q 1. - Calculus

Question

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) A limit exists if there is some real number that it is equal to.

(b) The limit of $f\left(x\right)$ as $x\to c$ is the value $f\left(c\right)$.

(c) The limit of $f\left(x\right)$ as $x\to c$ might exist even if the value of $f\left(c\right)$ does not.

(d) The two-sided limit of $f\left(x\right)$ as $x\to c$ exists if and only if the left and right limits of $f\left(x\right)$ exists as $x\to c$.

(e) If the graph of $f$ has a vertical asymptote at $x=5$, then $\underset{x\to 5}{\lim }f\left(x\right)=\infty$.

(f) If $\underset{x\to 5}{\lim }f\left(x\right)=\infty$, then the graph of $f$ has a vertical asymptote at $x=5$.

(g) If $\underset{x\to 2}{\lim }f\left(x\right)=\infty$, then the graph of $f$ has a horizontal asymptote at $x=2$.

(h) If $\underset{x\to \infty }{\lim }f\left(x\right)=2$, then the graph of $f$ has a horizontal asymptote at $y=2$.

Step-by-Step Solution

Answer

(a) The given statement is true.

(b) The given statement is false.

(c) The given statement is true.

(d) The given statement is true.

(e) The given statement is false.

(f) The given statement is true.

(g) The given statement is false.

(h) The given statement is true.

Part (a) step 1. Given information.

A limit exists if there is some real number that it is equal to.

Part (a) Step 2. Determine the given statement is true or false.

We know that the limit expression is given by $\underset{x\to c}{\lim }f\left(x\right)=L$.

The limit exists if there is some real number that it is equal to.

Hence, the statement is true.

Part (b) step 1. Given information.

The limit of $f\left(x\right)$ as $x\to c$ is the value $f\left(c\right)$.

Part (b) Step 2. Determine the given statement is true or false.

From the limit expression $\underset{x\to c}{\lim }f\left(x\right)$, the value is not $f\left(c\right)$, but it is almost $f\left(c\right)$ excluding the value at $x=c$.

Hence, the given statement is false.

Part (c) step 1. Given information.

The limit of $f\left(x\right)$ as $x\to c$ might exist even if the value of $f\left(c\right)$ does not.

Part (c) Step 2. Determine the given statement is true or false.

From the limit expression $\underset{x\to c}{\lim }f\left(x\right)$, the limit might exist even if the value $f\left(c\right)$ does not exist.

Hence, the statement is true.

Part (d) step 1. Given information.

The two sided limit of $f\left(x\right)$ as $x\to c$ exists if and only if the left and right limits of $f\left(x\right)$ exists as $x\to c$.

Part (d) Step 2. Determine the given statement is true or false.

From the graph, $x$ approaches $-1$ from the left side the height of the graph approaches $y=1$$\underset{x\to -1}{\lim }f\left(x\right)=1$.

If $x$ approaches $-1$from the left side the height of the graph approaches $y=1$, $\underset{x\to -1}{\lim }f\left(x\right)=1$.

If $\underset{x\to -1}{\lim }f\left(x\right)=\underset{x\to -1}{\lim }f\left(x\right)$, then $\underset{x\to -1}{\lim }f\left(x\right)=1$.

Hence, the given statement is true.

Part (e) step 1. Given information.

If the graph of $f$ has a vertical asymptote at $x=5$, then $\underset{x\to 5}{\lim }f\left(x\right)=\infty$.

Part (e) Step 2. Determine the given statement is true or false.

From the graph, $f$ has a vertical asymptote, then either $\underset{x\to 5}{\lim }f\left(x\right)=\infty$ or $\underset{x\to 5}{\lim }f\left(x\right)=-\infty$.

Hence, the given statement is false.

Part (f) step 1. Given information.

If $\underset{x\to 5}{\lim }f\left(x\right)=\infty$, then the graph of $f$ has a vertical asymptote at $x=5$.

Part (f) Step 2. Determine the given statement is true or false.

If the limit expression is given by $\underset{x\to 5}{\lim }f\left(x\right)=\infty$ or $\underset{x\to 5}{\lim }f\left(x\right)=-\infty$, the graph must have a vertical asymptote at $x=5$.

Hence, the given statement is true.

Part (g) step 1. Given information.

If $\underset{x\to 2}{\lim }f\left(x\right)=\infty$, then the graph of $f$ has a horizontal asymptote at $x=2$.

Part (g) Step 2. Determine the given statement is true or false.

If the limit expression is given by $\underset{x\to 2}{\lim }f\left(x\right)=\infty$, the graph must have a vertical asymptote at $x=2$ and no horizontal asymptote.

Hence, the statement is false.

Part (h) Step 1. Given information.

If $\underset{x\to \infty }{\lim }f\left(x\right)=2$, then the graph of $f$ has a horizontal asymptote at $y=2$.

Part (g) Step 2. Determine the given statement is true or false.

If the limit expression is given by $\underset{x\to \infty }{\lim }f\left(x\right)=2$, the graph would have a horizontal asymptote at $y=2$.

Hence, the given statement is true.