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Chapter 1: Q. 53 (page 98)

For each limit limx鈫�cf(x)=Lin Exercises 43鈥�54, use graphs and algebra to approximate the largest value of 未such that if localid="1648023101818" x鈭�(c-未,c)鈭�(c,c+未)thenf(x)鈭�(L-蔚,L+蔚)

localid="1648023199049" role="math" limx鈫�-1x2-2x-3x+1=-4,蔚=1

Short Answer

Expert verified

The required value of未=1

Step by step solution

01

Step 1. Given Information

The given expression islimx鈫�-1x2-2x-3x+1=-4,蔚=1

02

Step 2. Explanation

From the given expression, we have, c=-1,L=-4

The limit expression can be written as a formal statement as below,

For all epsilon positive, there exists a delta positive such that if localid="1648023264719" x鈭�(-1-未,-1)鈭�(-1,-1+未)Thenx2-2x-3x+1鈭�(-4-蔚,-4+蔚)

Now, the largest value of delta is given by,

localid="1648023296869" x2-2x-3x+1=L+蔚x2-2x-3x+1=-4+1x-3=-3x=0

Thus,

未=0+1未=1

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Most popular questions from this chapter

Sketch a labeled graph of a function that fails to satisfy the hypothesis of the Intermediate Value Theorem, and illustrate on your graph that the conclusion of the Intermediate Value Theorem does not necessarily hold.

Use the delta-epsilon definition of continuity to argue that f is or is not continuous at the indicated point x=c.

f(x)=2鈭�x,鈥呪赌呪赌呪赌�ifxrationalx2,鈥呪赌呪赌呪赌�ifxirrational,c=2

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) A limit exists if there is some real number that it is equal to.

(b) The limit of fxas x鈫�cis the value fc.

(c) The limit of fxas x鈫�cmight exist even if the value of fcdoes not.

(d) The two-sided limit of fxas x鈫�cexists if and only if the left and right limits of fxexists as x鈫�c.

(e) If the graph of fhas a vertical asymptote at x=5, then limx鈫�5fx=鈭�.

(f) If limx鈫�5fx=鈭�, then the graph of fhas a vertical asymptote at x=5.

(g) If limx鈫�2fx=鈭�, then the graph of fhas a horizontal asymptote at x=2.

(h) Iflimx鈫�鈭�fx=2, then the graph offhas a horizontal asymptote aty=2.

Write delta-epsilon proofs for each of the limit statements in Exercises 47鈥�60.

limx鈫�5+x-5=0

State what it means for a function f to be left continuous at a point x = c, in terms of the delta鈥揺psilon definition of limit.
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