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.

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

(g) True. If limx→6f(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.

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