# 数学代写|微积分代写Calculus代写|MATH1111

## 数学代写|微积分代写Calculus代写|Finding limits graphically

In section $1.4$ we evaluated limits “algebraically” and then considered what the graph might look like. Given the limit, we sketched a snippet of graph. The reverse can also be done: from the graph, we should be able to determine the limit. We call this determining a limit graphically. Knowing which limits give rise to holes, jumps, and vertical asymptotes, when we see such features on a graph, we can identify the limit accordingly.
Example 4 Use figure 3 to determine the following limits:

Another way of seeing this limit is to ask: As $x$ nears 2, what happens to the $y$-coordinates on the graph? In figure 5 , the red arrows on the horizontal axis point to $x=2$. Follow the curve above these red arrows; the result is in green. Then, go from the green arrows to the vertical axis, where the resulting arrows are magenta. We conclude that as $x$ nears $2, f(x)$ nears 1 . Therefore, $\lim {x \rightarrow 2} f(x)=1$. (b) To determine $\lim {x \rightarrow 4} f(x)$, we look at the graph near $x=4$, as in figure 6 . We see a vertical asymptote that exits the top of the picture on both sides of $x=4$. This matches the picture in section $1.4$ figure 13 , and we conclude that $\lim {x \rightarrow 4} f(x)=\infty$. The implication in the picture given by the dotted line representing the vertical asymptote is that near $x=4$, the values of $f(x)$ become infinitely large. Using arrows instead, we see that the magenta arrow in figure 6 is pointing, not to a specific real number, but rather off the top of the diagram to an infinite height, so that the limit is $\infty$. (c) For $\lim {x \rightarrow 6} f(x)$, we look near $x=6$, as in figure 7. We see a jump in the graph, which means the limits from the left and the right are not equal. We conclude that $\lim {x \rightarrow 6} f(x)$ DNE. Compare to section $1.4$ figure 11. Using arrows instead, note that the two magenta arrows in figure 7 do not point to the same number, so the rwo-sided limit does not exist. (d) To determine a one-sided limit such as $\lim {x \rightarrow 6^{+}} f(x)$, we only wish to look at one side, not both sides. Instead of looking at figure 7, we look at figure 8 , which shows the graph only on the righthand side of $x=6$. As always, the value of the function at $x=6$ is irrelevant, so it does not matter which circle is filled in and which is not. Instead, we pay attention to the curve and where it is headed, which is toward the $y$-coordinate 0 . We conclude that $\lim _{x \rightarrow 6^{+}}$ $f(x)=0$

## 数学代写|微积分代写Calculus代写|Sketching functions from limit information

We have previously sketched snippets of the graph of a function from information provided by limits. To practice the connection between graphs and limits even further, we can give several limits and sketch the graph of a function that meets all the given information. The general idea is to sketch the graph snippets as before, all on one graph, and then connect the snippets, yielding the graph of a function.

Example 5 Sketch the graph of a function f satisfing $\lim {x \rightarrow 2^{-}} f(x)=1$, $\lim {x \rightarrow 2^{+}} f(x)=3, \lim _{x \rightarrow-1} f(x)=-1$, and $f(2)=0$.

Solution We begin by drawing the graph snippets corresponding to each of the limits. For $\lim {x \rightarrow 2^{-}} f(x)=1$, we draw an open circle at the point $(2,1)$ with a snippet of graph on the left side. For $\lim {x \rightarrow 2^{+}}$ $f(x)=3$, we draw an open circle at the point $(2,3)$ with a snippet of graph on the right side. For $\lim _{x \rightarrow-1} f(x)=-1$, we draw an open circle at the point $(-1,-1)$ with a snippet of graph on both sides. Last, we have one more piece of information-namely, $f(2)=0$. Just as when plotting points in algebra, we place a point on the graph at the point $(2,0)$. These steps are shown in figure 13.

Now all that is left is to connect the pieces of graph. This can be done in any manner one wishes, as long as the result is a function, and the required limits and function values are preserved. Two of the infinitely many possible answers are in figure 14.

It is apparent that just knowing a few limits on the graph of a function is not enough to specify the function completely. Thankfully, there are additional tools of calculus that can produce information to shape a function’s graph. Stay tuned!

# 微积分代考

## 数学代写|微积分代写Calculus代写|Sketching functions from limit information

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