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 $\lim _{x\rightarrow 1} \sqrt{x+1}$.