Definition of limitsLimits are a tool that lets us understand what shouldhappen to a function at a x = a by using information aboutwhat is happening to the function near x = a. In otherwords, we say limx→af(x) = L if based on what the functionis doing close to x = a the value of the function at x = ashould be L.The key is understanding “close”; this is a measurement ofdistance. For numbers x and y we measure the distancebetween them by |x − y| (in general it is useful to think ofabsolute value as a representation of distance).Note that in particular we have two things that are gettingclose, namely the input has x getting close to a and theoutput has f(x) getting close to L. So we have to be able tofind the right blend of how to put these two notionstogether. One motivating idea to think of is that we alwayshave the ability to control the inputs and so the realchallenge is to figure out if it is possible to control theoutputs by our choice of what we do with the inputs.The technical definition of limits is as follows:We say limx→af(x) = L if for any ε > 0 there is aδ > 0 so that if 0 < |x − a| < δ then |f(x) − L| < ε.Here ε and δ are (small) bounds on how far apart thingscan be. So |f(x) − L| < ε means that f(x) and L are at mostε apart. Similarly x and a are at most δ apart. The reasonwe have 0 < |x − a| is to rule out looking at x = a;remember, limits are what should happen and not whatdoes happen, so we need to ignore what happens at a.So to show that a limit exists we play the following“game”: give us an ε and we will show you how to find aδ to satisfy the definition above.One of the most useful tools in playing this game is thetriangle inequality:|x + y| 6 |x| + |y| .Useful approaches to establish limits:• We can always make |x − a| small; so try and find|x − a| to show up somewhere in the expression andthen make it “small enough”.• Use the triangle inequality and break into parts andmake each piece small (some fractional part of ε.• When we have products look to make one thing smalland the other bounded.• We can combine a (finite) collection of constraints of δby taking the minimum.• Do not try to get the best δ; we just need a δ thatworks.The “knack” at getting good at working with limits isfiguring out what is important (in particular what can bemade small); what can be bounded (not arbitrarily large);and what can be ignored. This comes with practice and isnot always easy. In general working with expressions withinequalities can be harder than an expression with anequality because there are more options to manipulate theexpression.Problems1. Use the definition of limit to show thatlimx→1(x2+ 4x + 3) = 8.2. Use the definition of limit to show that limx→21x=12.3. Is the following statement True or False?If limx→af(x) = L and limx→af(x) = M, then L = M.4. Use the definition of limits to show that if limx→af(x) = Land limx→ag(x) = M, then limx→af(x) + g(x)= L + M.5. Use the definition of limits to show that if limx→af(x) = Land limx→ag(x) = M, then limx→af(x)g(x)=