# 1.8 | Limits

We write $$\displaystyle \lim_{x \rightarrow c} f(x) = L$$ if the values of $$f(x)$$ approach $$L$$ as $$x$$ approaches $$c$$.
A function $$f$$ is defined on an interval around $$c$$, except perhaps at the point $$x = c$$. We define the limit of the function $$f(x)$$ as $$x$$ approaches $$c$$, written $$\displaystyle \lim_{x \rightarrow c} f(x)$$, to be a number $$L$$ (if one exists) such that $$f(x)$$ is as close to $$L$$ as we want whenever $$x$$ is sufficiently close to $$c$$ (but $$x \not = c$$. If $$L$$ exists, we write
$$\lim_{x \rightarrow c} f(x) = L.$$
We define $$\displaystyle \lim_{x \rightarrow c} f(x)$$ to be the number $$L$$ (if one exists) such that for every $$\epsilon >0$$ (as small as we want), there is a $$\delta >0$$ (sufficiently small) such that $$|x-c| < \delta$$ and $$x \not = c$$, then $$|f(x) - L| < \epsilon$$.
Properties of Limits:
Assuming all the limits on the right-hand side exists:

1. If $$b$$ is a constant, then $$\displaystyle \lim_{x \rightarrow c} \left( b f(x) \right) = b \left( \lim_{x \rightarrow c} f(x) \right)$$.
2. $$\displaystyle \lim_{x \rightarrow c} \left( f(x) + g(x) \right) = \lim_{x \rightarrow c} f(x)+\lim_{x \rightarrow c} g(x)$$.
3. $$\displaystyle \lim_{x \rightarrow c} \left( f(x) g(x) \right) = \left( \lim_{x \rightarrow c} f(x) \right) \left(\lim_{x \rightarrow c} g(x) \right)$$.
4. $$\displaystyle \lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow c} f(x)}{\lim_{x \rightarrow c} g(x)}$$, provided $$\displaystyle \lim_{x \rightarrow c} g(x) \not = 0$$.
5. For any constant $$k$$, $$\displaystyle \lim_{x \rightarrow c} k = k$$.
6. $$\displaystyle \lim_{x \rightarrow c} x = c$$.
If $$f(x)$$ gets as close to a number $$L$$ as we please when $$x$$ gets sufficiently large, then we write
$$\lim_{x \rightarrow \infty} f(x) = L.$$
Similarly, if $$f(x)$$ approaches $$L$$ when $$x$$ is negative and has a sufficiently large absolute value, then we write
$$\lim_{x \rightarrow – \infty} f(x) = L.$$
The function $$f$$ is continuous at $$x = c$$ if $$f$$ is defined at $$x = c$$ and if
$$\lim_{x \rightarrow c} f(x) = f(c).$$
In other words, $$f(x)$$ is as close as we want to $$f(c)$$ provided $$x$$ is close enough to $$c$$. The function is continuous on the interval $$[a,b]$$ if it is continuous at every point of the interval. If $$c$$ is an endpoint of the interval, we define continuity at $$x=c$$ using one-sided limits at $$c$$.
Suppose that $$f$$ and $$g$$ are continuous on an interval and the $$b$$ is a constant. Then, on that same interval.

1. $$bf(x)$$ is continuous.
2. $$f(x) + g(x)$$ is continuous.
3. $$f(x)g(x)$$ is continuous.
4. $$f(x)/g(x)$$ is continuous, provided $$g(x) \not = 0$$ on the interval.
If $$f$$ and $$g$$ are continuous, and if the composite function $$f(g(x))$$ is defined on an interval, then $$f(g(x))$$ is continuous on that interval.