# [SOLVED]Simple question about Limit Properties

#### Cbarker1

##### Active member

$$\displaystyle lim f(bx)= b*lim f(x)$$ as x approach c

Thank you

Cbarker1

#### Guest

##### Active member
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

##### Well-known member
Hello, Cbarker1!

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: .$$3\cdot\lim_{x\to1}f(x) \:=\:3\cdot\lim_{x\to1}(2x+7) \:=\:3\cdot 9 \:=\:27$$

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