Limits from a Graph The graphs of f and g are given. Use...

Limits from a Graph The graphs of $f$ and $g$ are given. Use them to evaluate each limit if it exists. If the limit does not exist, explain why. $$\begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 2^{2}}[f(x)+g(x)]} & {\text { (b) } \lim _{x \rightarrow 1}[f(x)+g(x)]} \\ {\text { (c) } \lim _{x \rightarrow 0}[f(x) g(x)]} & {\text { (d) } \lim _{x \rightarrow-1} \frac{f(x)}{g(x)}} \\ {\text { (e) } \lim _{x \rightarrow 2} x^{3} f(x)} & {\text { (f) } \lim _{x \rightarrow 1} \sqrt{3+f(x)}}\end{array}$$

Limits from a Graph The graphs of $f$ and $g$ are given. Use them to evaluate each limit if it exists. If the limit does not exist, explain why.
$$\begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 2^{2}}[f(x)+g(x)]} & {\text { (b) } \lim _{x \rightarrow 1}[f(x)+g(x)]} \\ {\text { (c) } \lim _{x \rightarrow 0}[f(x) g(x)]} & {\text { (d) } \lim _{x \rightarrow-1} \frac{f(x)}{g(x)}} \\ {\text { (e) } \lim _{x \rightarrow 2} x^{3} f(x)} & {\text { (f) } \lim _{x \rightarrow 1} \sqrt{3+f(x)}}\end{array}$$

Key Concepts

Discontinuities

Discontinuities refer to points at which a function is not continuous—its graph may have holes, jumps, or infinite breaks. Understanding the nature of discontinuities is crucial when evaluating limits from a graph, as the existence of different one-sided limits or undefined values at the point might mean that the limit does not exist even if each one-sided approach has a value.

Composite Functions in Limits

Composite functions involve applying one function to the result of another, and their limits can typically be evaluated using the limit of the inner function first, followed by the outer function, provided the outer function is continuous at the limit of the inner function. This concept is particularly relevant when dealing with expressions such as square roots or polynomial multiples of a function value.

Limit Laws

Limit laws provide a set of rules that allow the evaluation of limits of complex functions by breaking them down into simpler parts. These include the sum, product, quotient, and composition rules, which state that if limits of individual functions exist, then the limit of their sum, product, or quotient (given the denominator's limit is not zero) can be found by the corresponding operation on the limits.

One-Sided Limits

One-sided limits are used when considering the behavior of a function as the input approaches a point from one side only—either from values greater than or less than the point of interest. They are particularly useful in cases where the function behaves differently on either side of the point, which may lead to the overall limit not existing if the two one-sided limits are not equal.

Graphical Analysis of Limits

Graphical analysis involves using the visual information from a function's graph to determine its limit as the input approaches a certain value. This method helps in identifying trends and behaviors such as approaching a horizontal line, identifying jumps, or determining if the limit fails to exist due to oscillations or discontinuities.

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a particular value. They capture what value, if any, the function is approaching, regardless of what the function actually does at that point. This concept is used to understand function behavior near points of interest, including points of discontinuity.

Transcript

00:01 Okay, this question asks us to evaluate various limits.

00:05 So part a wants the limit as x approaches 2 of f of x plus g of x.

00:15 So we can just use our limit properties here and split this up into the sum of two limits.

00:29 And now we just read these limits from the graph.

00:32 So x approaches 2 of f of x if we look here well that's going to go to two on both sides and then g of x as we approach 2 we go to 0 so part a is 2 then next up for part b we have the same thing except now we're approaching 1 so again we'll just distribute this limit to each of the functions and add them together and now we'll now this time we'll read what's the graph doing near x equals 1.

01:36 Well for f of x it's fine, it's just approaching 1, but g of x it does not exist.

01:50 So this whole overall limit does not exist because you can only add them together like this if each of the limits by themselves exist and they don't.

02:06 So this overall limit doesn't make sense.

02:09 So part c wants the limit of the product of these two functions.

02:19 And again, we do the same thing as before, where we just take the limits of each function and then multiply them.

02:35 So now reading these off the graph, this is just, well, as we approach 0 and f of x, we get to 0.

02:43 And as we approach 0 for g of x, i'd estimate that to be about 1 .5.

02:58 But it doesn't matter because regardless it's 0.

03:05 Then for part d, it wants the limit as x approaches negative 1 of f of x divided by g of x.

03:19 So again, we just divide the limits.

03:37 So doing this, as we approach negative 1, f of x is approaching negative 1...

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