Simple question about Limit Properties

[cbarker1]

I read a textbook about limits. I saw several properties about the limits.

\(\displaystyle lim f(bx)= b*lim f(x)\) as x approach cThank you

Cbarker1

 

[Guest2]

Pick $ f(y) = \frac{1}{y}$. Then $\displaystyle \lim_{x \to c} f(bx) = \lim_{x \to c}\frac{1}{bx} = \frac{1}{bc}$. On the other hand $\displaystyle \lim_{x \to c} bf(x) = b\lim_{x \to c}\frac{1}{x} = \frac{b}{c}.$

Of course $ \frac{1}{bc} \ne \frac{b}{c}$. I'm missing something or the property is false (or they meant something else).

 

[soroban]

Hello, Cbarker1!

I read a textbook about limits.
I saw several properties about the limits.

\(\displaystyle \lim_{x\to c} f(bx)\:=\: b\cdot\lim_{x\to c}f(x)\)

This is not true.

Let $$f(x) \:=\:2x+7$$

Then: .$$\lim_{x\to1}f(3x) \:=\:\lim_{x\to1}(6x+7) \:=\:13$$

But: .$$3\cdot\lim_{x\to1}f(x) \:=\:3\cdot\lim_{x\to1}(2x+7) \:=\:3\cdot 9 \:=\:27$$Perhaps you misread the identity.
The following is true.

. . $$\lim_{x\to c}b\!\cdot\!\!f(x) \;=\;b\!\cdot\!\lim_{x\to c}f(x)$$