Question

Suppose that f(x) denotes a function defined for all real numbers. Each of the statements below us true "sometimes". Give an example for each function to prove it holds a true and false function.

Answer 1

\[\lim_{x \rightarrow -7} f(x) =a\] then
\[\lim_{x \rightarrow -7}\frac{ 1 }{ f(x) } = \frac{ 1 }{ a }\]
You may choose any value of 'a' that you wiah for your examples.

Answer 2

@DebbieG

Answer 3

Well, starting with the false example... I think if you pick a function so that the first limit is =0 (think simple, lol... a linear function with f(-7)=0 will do, so can you think of how to "cook up" that function?)...... then the second limit won't even exist, much less be =1/a, because the function will "blow up" to +- infinity, coming from either side of x=-7.

To make it true.... the simplest thing would be a constant function, dontcha think? :) that is, if f(x) is just a horizontal line y=a for any non-zero value of a, then I believe it will be true.

Although that's a reallllly boring example, lol. Really, any simple function will do if it is continuous and non-zero at x=-7. So a quadratic.... something like.... f(x)=(x+8)^2, is a "snazzier" example than the constant function. But either will do. :)