# Understanding Limits: Evaluating Tricky Expressions

• johnq2k7
In summary, for problem a, the limit does not exist because the numerator and denominator approach different values as x approaches infinity. For problem b, the limit also does not exist because the quotient approaches infinity without bound as x approaches pi.
johnq2k7
2.) Evaluate the following limits, justifying your answers. If a limit does not exist explain why.

a.) lim (x--> inf.) (3x^3 +cos x)/(sin x- x^3)

b.) lim (x-->Pie(+)) (tan^-1 (1/(x-Pie)))/(Pie-x

For a, factor x^3 from both terms in the numerator and both terms in the denominator.
For b, if I'm interpreting what you wrote correctly, the numerator is approaching pi/2 (note spelling -- the name of this Greek letter is pi, not pie), and the denominator is approaching 0, so the quotient is getting large without bound.

## 1. What is a limit?

A limit is the value that a function approaches as the input approaches a certain value or as the output approaches a certain value. It is used to describe the behavior of a function near a specific point.

## 2. How do I solve limits?

To solve a limit, you can use a variety of techniques such as direct substitution, factoring, rationalizing, and using trigonometric identities. It is important to understand the properties of limits and to follow the appropriate steps for each type of limit.

## 3. What is the difference between one-sided and two-sided limits?

A one-sided limit only considers the behavior of a function approaching a specific value from one side, either the left or the right. A two-sided limit considers the behavior from both sides and requires that the function approaches the same value from both sides in order for the limit to exist.

## 4. Can limits be undefined?

Yes, a limit can be undefined if the function approaches different values from the left and right sides, if there are infinite oscillations, or if the function has a vertical asymptote at the specific value.

## 5. Why are limits important in mathematics and science?

Limits are important because they allow us to understand and describe the behavior of functions as they approach certain values. They are used in various fields of mathematics and science, such as calculus, physics, and statistics, to model real-world situations and make predictions about their behavior.

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