Using the Precise definition to prove the Constant Multiple Rule

[sad.ib.kid]

Hi folks! Me and all my other classmates are struggling with this one problem on the homework. I do understand the precise definition and the constant multiple rule and am comfortable with solving problems that involve either one of them. However, it racks my brain when the two concepts are put into one question and I am stumped. Thank you in advance if you choose to help me, I really appreciate it as a stressed college student <3

 

[pka]

Hi folks! Me and all my other classmates are struggling with this one problem on the homework. I do understand the precise definition and the constant multiple rule and am comfortable with solving problems that involve either one of them. However, it racks my brain when the two concepts are put into one question and I am stumped. Thank you in advance if you choose to help me, I really appreciate it as a stressed college student <3 View attachment 22008

Suppose that \(\varepsilon>0\) by definition \((\exists\delta>0)\)\([|x-c|<\delta \Rightarrow |L-f(x)|<\varepsilon \)
Note that \(\large |K\cdot L-K\cdot f(x)|=|K|\cdot |L-f(x)|\) How would you finish?

 

[sad.ib.kid]

Suppose that \(\varepsilon>0\) by definition \((\exists\delta>0)\)\([|x-c|<\delta \Rightarrow |L-f(x)|<\varepsilon \)
Note that \(\large |K\cdot L-K\cdot f(x)|=|K|\cdot |L-f(x)|\) How would you finish?

My first instinct is to divide ε by K to balance out the inequality, but when I do that, I am not sure how I would implement that to find the open interval

 

[pka]

My first instinct is to divide ε by K to balance out the inequality, but when I do that, I am not sure how I would implement that to find the open interval

Why not find a \(\delta'\) that forces \(|L-f(x)|<\dfrac{\varepsilon}{|L|+1}~?\)