Enroll in one of our FREE online STEM bootcamps. Join today and start acing your classes!View Bootcamps

MM

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75
Problem 76
Problem 77
Problem 78
Problem 79
Problem 80
Problem 81
Problem 82
Problem 83
Problem 84
Problem 85
Problem 86

Problem 48

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .

$$\lim _{x \rightarrow 3}\left(\frac{3}{4} x+1\right)$$

Answer

Click for the solution and explanation.

You must be logged in to like a video.

You must be logged in to bookmark a video.

## Discussion

## Video Transcript

and this problem, we want to find the limit as XO. Purchase three of 3/4 x plus one one. Use that slime Delta definition of limit to find this limit. In other words, you want to show that for every up song grease zero, there exists a delta greater than zero such that the function through X over four plus one, minus 13 over four absolute value is less than epsilon. Whenever X minus three, absolute value is between zero and Delta. No notice that are proposed to limit is 13 over four. This is because for linear functions, which have the form F of X equals MX plus B, the limit as X approaches, si is always going to be, and C equals B, meaning we can just plug in this value were approaching into the function. So also note that if we plug that value in for X, we will get 13 over four. So that's kind of what we proposed that limit. But we need to show this because we're assuming we don't know this claim to be good with. So look at this value here again. So notice that when we simplify this by distributing this negative sign and combining the water Negative three thirteen 13 over four will get three X over four, minus nine over. For I noticed there we can kind of factor out 3/4. So who had the absolute value of 3/4 times X minus tree? We want to factor that out because we want to establish a relationship between this expression and this bound that we have a next my history or down that we have on the absolute value of X fine story. So notice that that term kind of appears here. Further, we can pull out the absolute value of three or four times absolutely of X minus three. And since three over four is already positive, the absolute value of that will just be three or four. So we have three or four times the absolute value of X, my history. So now it seems we should let Delta bi equal to Absalon, which ought to know by the letter a absolute divided by three over fourths, which is the same as four times up Salon over three, and we're done