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Let$ f(x) = \left\{ \begin{array}{ll} x^2 + 1 & \mbox{if $ x < 1 $}\\ (x - 2)^2 & \mbox{if $ x \ge 1 $} \end{array} \right.$

(a) Find $ \displaystyle \lim_{x \to 1^-}f(x) $ and $ \displaystyle \lim_{x \to 1^+}f(x) $.(b) Does $ \displaystyle \lim_{x \to 1}f(x) $ exist?(c) Sketch the graph of $ f $.

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

Campbell University

Oregon State University

Baylor University

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 problem Number fifty with the sewer calculus Safe Edition section two point three Let f equal and this is a piece of ice function. So the first function is X squared, plus one if X is less than one and the quantity X minus two squared If X is greater than or equal to one party find the limit is exporters one from the left of F and the limit his expertise one from the right of f the limit as experts is one from the left of we'LL have to do with dysfunction. Since we're approaching one from the left, this only applies to this region. X is less than one, and so our function half will be X squared plus one and through directs an institution we played in one square two plus one gives us our value of two for the first limit for the second limit. The limit, his expressions that one from the right. We are in this region greater than or equal to one, which means that our function is Dequan titty X minus two squared and through direct institution one minutes Tuesday at one. It's quantity squared is positive. One. Prepare B does limit his expertise. One of enough exists. It does not exist because two does not equal one. The left limit is not equal to right. Lim. Therefore, this limit does not exist and finally for part. See, we need This gets aggressive. The function f we need to plot the first function for this region X is less than one and then plot. The second function, X minus two Quantity squared for X is greater than equal to one. Here's an example of ah plotting mechanism where the ranges that don't mean is restricted for each function. Hex QUOTABLES one is a parabola and as we can see as it purchased a positive one, that is where we a counter a hole. And then he jumped down to the next function explains to quantity squared where that continue function continues on after X is equal to one Here. We also see that the limit does not exist. That X equals two because the function approaches a different value two from the left than it does from the right one

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