### Elementary limits

The idea of **limit** is intended to be merely a slight extension of
our *intuition*. The so-called $\varepsilon,\delta$-definition was
invented after people had been doing calculus for hundreds of years,
in response to certain relatively pathological technical
difficulties. For quite a while, we will be entirely concerned with
situations in which we can either ‘directly’ see the value of a limit
*by plugging the limit value in*, or where we *transform* the
expression into one where we *can* just plug in.

So long as we are dealing with functions no more complicated
than polynomials, most *limits* are easy to understand: for
example,
$$\lim_{x\rightarrow 3} 4x^2+3x-7= 4\cdot (3)^2+3\cdot(3)-7=38$$
$$\lim_{x\rightarrow 3} {4x^2+3x-7\over 2-x^2}=
{4\cdot (3)^2+3\cdot(3)-7\over 2-(3)^2}={38\over -7}$$.

The point is that we just substituted the ‘$3$’ in and *nothing
bad happened*. This is the way people evaluated easy limits for
hundreds of years, and should always be the first thing a person does,
just to see what happens.

#### Exercises

- Find $\lim _{x\rightarrow 5} 2x^2-3x+4$.
- Find $\lim _{x\rightarrow 2} { x+1 \over x^2+3 }$.
- Find $\lim _{x\rightarrow 1} \sqrt{x+1}$.