Q. 1 - Calculus

Question

True/False: 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) True or False: For $\underset{x\to c}{\lim }f\left(x\right)$ to be defined, the function f must be defined at x = c.

(b) True or False: We can calculate a limit of the form $\underset{x\to c}{\lim }f\left(x\right)$ simply by finding f(c).

(c) True or False: If $\underset{x\to c}{\lim }f\left(x\right)=10,$ then f(c) = 10.

(d) True or False: If f(c) = 10, then $\underset{x\to c}{\lim }f\left(x\right)=10.$

(e) True or False: A function can approach more than one limit as x approaches c.

(f) True or False: If $\underset{x\to 4}{\lim }f\left(x\right)=10,$ then we can make f(x) as close to 4 as we like by choosing values of x sufficiently close to 10.

(g) True or False: If $\underset{x\to 6}{\lim }f\left(x\right)=\infty ,$ then we can make f(x) as large as we like by choosing values of x sufficiently close to 6.

(h) True or False: If $\underset{x\to \infty }{\lim }f\left(x\right)=100,$ then we can find values of f(x) between 99.9 and 100.1 by choosing values of x that are sufficiently large.

Step-by-Step Solution

Answer

Part (a) The given statement is false.

Part (b) The given statement is false. 

Part (c) The given statement is false. 

Part (d) The given statement is false. 

Part (e) The given statement is false. 

Part (f) The given statement is false. 

Part (g) The given statement is True. 

Part (h) The given statement is True. 

Part (a) Step 1. Given Information.

The given limit is $\underset{x\to c}{\lim }f\left(x\right).$

Part (a) Step 2. Determining whether the statement is true or false.

The given statement "For $\underset{x\to c}{\lim }f\left(x\right)$ to be defined, the function f must be defined at x = c"  is  false because the limit can be defined at any value of x=c.

Part (b) Step 1. Determining whether the statement is true or false.

The given statement "We can calculate a limit of the form $\underset{x\to c}{\lim }f\left(x\right)$ simply by finding f(c)"  is false because for the limit $\underset{x\to c}{\lim }f\left(x\right)$ the value is not just equal to f(c).

Part (c) Step 1. Determining whether the statement is true or false.

The given statement "If $\underset{x\to c}{\lim }f\left(x\right)=10,$ then f(c) = 10"  is false because for the limit $\underset{x\to c}{\lim }f\left(x\right)=10,$ then $f\left(c\right)\ne 10.$

Part (d) Step 1. Determining whether the statement is true or false.

The given statement " If f(c) = 10, then $\underset{x\to c}{\lim }f\left(x\right)=10$" is false because for the limit $\underset{x\to c}{\lim }f\left(x\right),$ the value is not equal to $f\left(c\right).$

Part (e) Step 1. Determining whether the statement is true or false.

The given statement "A function can approach more than one limit as x approaches c"  is false because if $\underset{x\to c}{\lim }f\left(x\right)=A$ and $\underset{x\to c}{\lim }f\left(x\right)=B$ then $A=B.$

Part (f) Step 1. Determining whether the statement is true or false.

The given statement "If $\underset{x\to 4}{\lim }f\left(x\right)=10,$ then we can make f(x) as close to 4 as we like by choosing values of x sufficiently close to 10"  is false.

Part (g) Step 1. Determining whether the statement is true or false.

The given statement "If $\underset{x\to 6}{\lim }f\left(x\right)=\infty ,$ then we can make f(x) as large as we like by choosing values of x sufficiently close to 6"  is true because the value of limit $x\to 6$ gives infinity.

Part (h) Step 1. Determining whether the statement is true or false.

The given statement "If $\underset{x\to \infty }{\lim }f\left(x\right)=100,$ then we can find values of f(x) between 99.9 and 100.1 by choosing values of x that are sufficiently large" is true.