Finding a value to make piecewise continuous

[BrianHare]

Homework Statement

Find c such that it makes f(x) continuous.

Homework Equations

<br /> f(x)=\begin{cases}<br /> 2x+c&amp;x &lt; -5\\<br /> 3x^2&amp;-5 \leq x &lt; 0\\<br /> cx^2&amp;0 \leq x\\<br /> \end{cases}<br />

The Attempt at a Solution

I know that
\lim_{x\to 5^-}3x^2 = 2x+c
and
\lim_{x\to 0^+}3x^2 = cx^2

Which makes the 2 points where it disconnects at (-5, 75) and (0,0)
Given that I make 2x+c = 75, where x=-5. I get C= 85, and since cx^2=0 is any real, does that mean the answer is c=85?

 

[tiny-tim]

Welcome to PF!

I know that
\lim_{x\to 5^-}3x^2 = 2x+c
and
\lim_{x\to 0^+}3x^2 = cx^2

Which makes the 2 points where it disconnects at (-5, 75) and (0,0)
Given that I make 2x+c = 75, where x=-5. I get C= 85, and since cx^2=0 is any real, does that mean the answer is c=85?

Hi Brian! Welcome to PF!

Yes, c = 85.

(though you have a strange way of using lim …

you might as well say lim 3x² = 3*(-5)², and so on. )

 

[BrianHare]

Hi Brian! Welcome to PF!

Yes, c = 85.

(though you have a strange way of using lim …

you might as well say lim 3x² = 3*(-5)², and so on. )

My teacher has a definition where

\lim_{x\to a}f(x) = f(c)

It was my understanding that f(x) = 3x^2, a = -5, and f(c) = 2x+c. So once I knew the answer to the limit, i knew that f(c) = 75, thus 2x+c must also be 75. Maybe I am misunderstanding the definition. Can anyone clarify?

 

[tiny-tim]

My teacher has a definition where

\lim_{x\to a}f(x) = f(c)

that doesn't make any sense …

what is f(c) supposed to mean?

(f(c) = 2c + c or 3c² or cc²)

does he mean \lim_{x\to a}f(x) = f(a)?​