# Is the limit of the slopes of lines equal to the slope of the limit?

Suppose that $$(a_n)$$ and $$(b_n)$$ are sequences of real numbers such that $$a_n x + b_n \rightarrow ax+b$$ for all $$x\in\mathbb{R}$$. Then my question is, is it necessarily true that $$a_n\rightarrow a$$? Or is it possible for $$(a_n)$$ to be a divergent sequence?

My question would be answered if a sequence of linear functions converges pointwise to a linear function if and only if it converges uniformly, but I’m not sure if that’s true. I’m guessing it is true.

I ask because this seems to be part of the intuition behind the idea that the slope of the tangent line is the limit of the slopes of the secant lines.

Taking $$x=0$$ we have $$b_{n}\rightarrow b$$. Taking $$x=1$$ we have $$a_{n}+b_{n}\rightarrow a+b$$, then by basic limit rule we have $$a_{n}\rightarrow a$$.
For uniform convergence, it is not. For $$f_{n}(x)=\dfrac{1}{n}x$$, $$f_{n}(x)\rightarrow 0$$ but $$\sup_{x}|f_{n}(x)|=\infty$$.
• Does $a_n x + b_n \rightarrow f(x)$ for all $x\in\mathbb{R}$ imply that $f$ is linear? Commented Nov 14, 2019 at 1:18
• So taking $x=0$ we have $b_{n}\rightarrow f(0)$. And taking $x=1$ we have $a_{n}+b_{n}\rightarrow f(1)$, by basic limit rule again we have $a_{n}\rightarrow f(1)-f(0)$, so with $x$ fixed, $f(x)=(f(1)-f(0))x+f(0)$. Commented Nov 14, 2019 at 1:22