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Use a graph to find a number $ N $ such that $$ \text{if} \hspace{5mm} x > N \hspace{5mm} \text{then} \hspace{5mm} \biggl| \frac{3x^2 + 1}{2x^2 + x + 1} - 1.5 \biggr| < 0.05 $$

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Campbell University

Oregon State University

Harvey Mudd College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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This is problem number seventy one of us to a calculus Eighth edition section to Prince six Is the graft to find a number in such that if X is greater than in, then the absolute value of the quantity three x worthless one, no matter where they wanted me to. X squared plus X plus one minus one point is less than zero point zero five. So we're going to use a graft. We're going to put this exact function here on the left side. This everything here within the emcee Valya, including now civility signs. We're going to compare it to the values zero point zero five. And if we do that here we have the function in the of the absolutely function in red. And we're comparing it with the y values zero point zero five. And if we would like this function Red Toby Ah, less than zero point zero five. Here we see that we must choose on and end value greater than or equal greater than fourteen point Tito for so in our case from our graph and is approximately fourteen point before. If fourteen point eight, if m is supposed to be an integer then when we would say is that thanks speaker to them. Fifteen. So fifteen is acceptable fourteen point game if it can be any number, is a bit more correct. Fourteen point oh four. But for our sake we'LL see the critics credit than fifteen guarantees that this and equality to the rate and satisfied.

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