High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ 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 [tex]\lim_{x \to c} f(x)[/tex] to be defined, the function [tex]f[/tex] must be defined at [tex]x = c[/tex].

(b) True or False: We can calculate a limit of the form [tex]\lim_{x \to c} f(x)[/tex] simply by finding [tex]f(c)[/tex].

(c) True or False: If [tex]\lim_{x \to c} f(x) = 10[/tex], then [tex]f(c) = 10[/tex].

(d) True or False: If [tex]f(c) = 10[/tex], then [tex]\lim_{x \to c} f(x) = 10[/tex].

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

(f) True or False: If [tex]\lim_{x \to 4} f(x) = 10[/tex], then we can make [tex]f(x)[/tex] as close to 4 as we like by choosing values of [tex]x[/tex] sufficiently close to 10.

(g) True or False: If [tex]\lim_{x \to 6} f(x) = \infty[/tex], then we can make [tex]f(x)[/tex] as large as we like by choosing values of [tex]x[/tex] sufficiently close to 6.

(h) True or False: If [tex]\lim_{x \to \infty} f(x) = 100[/tex], then we can find values of [tex]f(x)[/tex] between 99.9 and 100.1 by choosing values of [tex]x[/tex] that are sufficiently large.

Answer :

Final answer:

This response clarifies the concept of limits in calculus, specifically addressing common misconceptions and explaining true and false statements by providing relevant examples.

Explanation:

This series of questions is all about the concept of limits in calculus.

(a) False. The limit of a function as x approaches c doesn't require f to be defined at x=c. Example: f(x)=(x^2-1)/(x-1) for x≠1.

(b) False. Sometimes you can find limx→c​f(x) by finding f(c) but not always. Example: f(x) = (x^2-4)/(x-2) for x≠2, f(2) is undefined but the limit as x approaches 2 is 4.

(c) False. The limit of a function as x approaches c may not be equal to f(c). For example: f(x) = x when x≠2 , f(2) = 10, but limx→2f(x) = 2.

(d) False. If f(c)=10 it doesn't mean that limx→c​f(x)=10. Take f(x) = x when x≠2, f(2)=10, but the limit as x approaches 2 is 2.

(e) False. A function can have at most two limits (from right and left) as x approaches a given point.

(f) False. If limx→4​f(x)=10, then we can make f(x) close to 10 by selecting x sufficiently close to 4, not 10.

(g) True. If limx→6​f(x)=[infinity], we can make f(x) become as large as we like by choosing values of x close to 6 which is the definition of limit approaching infinity.

(h) True. If limx→[infinity]​f(x)=100, we can find values of f(x) close to 100 by selecting sufficiently large x-values.

Learn more about Limits in Calculus here:

https://brainly.com/question/35073377

#SPJ11