Question

Calculus questions!
I have three questions regarding limits and continuous functions

I will just need a minute to type the equations in!

Answer 1

\[\lim_{x \rightarrow 1} \frac{ x^3-1 }{ x^2-1 }\]

Answer 2

2.) \[\lim_{t \rightarrow 9} \frac{ 9-t }{ 3-\sqrt{t}}\]

Answer 3

the first one is it x^3-1/x^2-1

Answer 4

Yes it is

Answer 5

3.) find the values of c so that f(x) is continuous on the entire real number line when \[f(x)=\left\{ x-2, x≤5 \right\} \left\{ cx-3, x>5 \right\}\]
*both equations are supposed to be the one bracket (like how a piecewise function is set up)

Answer 6

i assume it is like that and continue. the discontinuity of that expresssion is when x=1
\(\Large \rm\color{midnightblue}{\lim_{x\to1}\frac{x^3-1}{x^2-1}=\lim_{x\to1}\frac{\cancel{(x-1)}(x^2+x+1)}{\cancel{(x-1)}(x+1)}=lim_{x\to1}\frac{x^2+x+1}{x+1}=\\lim_{x\to1}\frac{1+1+1}{1+1}=3/2}\)

Answer 7

the second multiply top and bottom by the conjugate \(3+\sqrt{t}\)

Answer 8

for 3 if f is continuous then the limit must exist and therefore
\(\Large \rm\color{midnightblue}{lim_{x\to5^-}f(x)=lim_{x\to5^+}f(x)}\)
set this equation and solve for c

Answer 9

Thanks for all your help!

Answer 10

Anytime! i hope you got the answers^_^

Answer 11

but i think you are okay with algebra :)