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Problem type

Limit of a function from the epsilon-delta definition

What it solves
Prove that a polynomial (a straight line, a quadratic or a cubic) has the limit you expect at a point, using only the epsilon-delta definition: for every output tolerance epsilon, find an input tolerance delta that guarantees it.
Subject
Real Analysis
Topic
Limits from the definition
Also called
epsilon delta, limit of a function, continuity from the definition
Your problem

Prove this limit using only the epsilon-delta definition, then try the proof out with the tolerance ε=110\varepsilon = \frac{1}{10}.

lim⁡x→−5(6x+7)=−23\lim_{x \to -5} \left( 6 x + 7 \right) = -23
Invent a new one:
Femtobytes
Epsilon-delta proof that the limit of 6x + 7 as x approaches -5 is -23, step by step
femtobytes.com/solve/epsilon-delta-limit/0_0_6_7_-5_1~10
The problem

Prove this limit using only the epsilon-delta definition, then try the proof out with the tolerance ε=110\varepsilon = \frac{1}{10}.

lim⁡x→−5(6x+7)=−23\lim_{x \to -5} \left( 6 x + 7 \right) = -23
The result
lim⁡x→−5(6x+7)=−23\lim_{x \to -5} \left( 6 x + 7 \right) = -23

Proved. Given any tolerance ε>0\varepsilon > 0, the choice δ=ε6\delta = \frac{\varepsilon}{6} works.

checked 9 lines of this solution were evaluated with exact arithmetic before it was shown to you. A solution with a false line is never served.
Picture 1 of 2Step 1: Find the candidate limit, then say exactly what must be proved
L + εL = -23L − εc − δc = -5c + δinput xoutput f(x)
The graph of the function near the point. The upright strip is the input window of half-width delta; the level band is the output tolerance of half-width epsilon. Inside the strip the graph never leaves the band. Here c=−5c = -5, L=−23L = -23, ε=110\varepsilon = \frac{1}{10} and δ=160\delta = \frac{1}{60}. The bright part of the graph is the part inside the strip. The hollow circle marks the point itself, which the definition leaves out.
1

Find the candidate limit, then say exactly what must be proved

Why this step

The definition can only CHECK a number we propose; it never hands us one. For a polynomial the natural candidate is the value at the point itself, because a polynomial is built from nothing but adding and multiplying, and neither of those can make a sudden jump. That is a belief. The proof turns it into a fact.

Put x=−5x = -5 into the function:

6(−5)+7\displaystyle 6 \left( -5 \right) + 7==−30+7\displaystyle -30 + 7
Do each multiplication. checked
==−23\displaystyle -23
Add the numbers up. checked
Claim: lim⁡x→−5(6x+7)=−23\text{Claim: } \lim_{x \to -5} \left( 6 x + 7 \right) = -23
This is what we will prove, with c=−5c = -5 and L=−23L = -23.
L + εL = -23L − εc − δc = -5c + δinput xoutput f(x)
The graph of the function near the point. The upright strip is the input window of half-width delta; the level band is the output tolerance of half-width epsilon. Inside the strip the graph never leaves the band. Here c=−5c = -5, L=−23L = -23, ε=110\varepsilon = \frac{1}{10} and δ=160\delta = \frac{1}{60}. The bright part of the graph is the part inside the strip. The hollow circle marks the point itself, which the definition leaves out.
2

Measure the distance between an output and the limit

Why this step

The definition is about one quantity only, ∣f(x)−L∣\left| f(x) - L \right|. We do not control it directly; we only control how close xx is to cc. So we rewrite the output distance until the input distance ∣x−c∣\left| x - c \right| appears inside it. Then controlling one controls the other.

∣6x+7−(−23)∣\displaystyle \left| 6 x + 7 - \left( -23 \right) \right|==∣6x+30∣\displaystyle \left| 6 x + 30 \right|
Combine the plain numbers: 7−(−23)=307 - \left( -23 \right) = 30. checked
==∣6(x−(−5))∣\displaystyle \left| 6 \left( x - \left( -5 \right) \right) \right|
Take out the common factor 66. Check by multiplying back: 6⋅5=306 \cdot 5 = 30. checked
==6∣x−(−5)∣\displaystyle 6 \left| x - \left( -5 \right) \right|
The absolute value of a product is the product of the absolute values, and ∣6∣=6\left| 6 \right| = 6. checked
L + εL = -23L − εc − δc = -5c + δinput xoutput f(x)
The graph of the function near the point. The upright strip is the input window of half-width delta; the level band is the output tolerance of half-width epsilon. Inside the strip the graph never leaves the band. Here c=−5c = -5, L=−23L = -23, ε=110\varepsilon = \frac{1}{10} and δ=160\delta = \frac{1}{60}. The bright part of the graph is the part inside the strip. The hollow circle marks the point itself, which the definition leaves out.
3

Rough work: how close must x be kept?

4

The proof, written forwards

5

Test the rule with a real tolerance