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## 数学代写|微积分代写Calculus代写|Continuity of polynomial and rational functions

The same type of proof that linear functions are continuous applies to quadratic functions as well.

Theorem 7 CONTINUTTY OF QUADRATIC FUNCTIONS If $f(x)=b x^2+c x+d$, then $f$ is continuous.

Proof. We wish to prove that $f$ is continuous at $x=k$ for every real number $k$. To do this, we need to show that $\lim {x \rightarrow k} f(x)=f(k)$. We start by determining $f(k)$ : $$f(k)=b k^2+c k+d$$ Next we check the limit: \begin{aligned} \lim {x \rightarrow k}\left(b x^2+c x+d\right) &=b(k+\alpha)^2+c(k+\alpha)+d \ &=b\left(k^2+2 k \alpha+\alpha^2\right)+c k+c \alpha+d \ &=b k^2+2 b k \alpha+b \alpha^2+c k+c \alpha+d \ & \approx b k^2+c k+d \end{aligned}
Because $\lim _{x \rightarrow k} f(x)=f(k), f$ is continuous at $x=k$. This is true for every real number $k_3$ so $f$ is continuous.

Parabolas can therefore be drawn using connect-the-dots. In a very similar manner, so can all polynomial functions. One proof that a polynomial of arbitrary degree is continuous is very similar to theorems 6 and 7. A different proof is presented at the end of this section.

## 数学代写|微积分代写Calculus代写|Where is f continuous?

Because elementary functions are continuous throughout their domains, asking where such a function is continuous is the exact same thing as asking where it is defined.
Example 1 On what intervals is $f(x)=\frac{2 x-9}{3 x-5}$ continuous?
Solution This is the exact same question as: What is the domain of $f(x)=\frac{2 x-9}{3 x-5}$ ? A rational function is defined everywhere except where we divide by zero, so we set the denominator equal to zero and solve:
\begin{aligned} 3 x-5 &=0 \ 3 x &=5 \ x &=\frac{5}{3} . \end{aligned}
A picture of the domain is given in figure 4. The domain of the function is $\left(-\infty, \frac{5}{3}\right) \cup\left(\frac{5}{3}, \infty\right)$. Our solution can be written as “the intervals $\left(-\infty, \frac{5}{3}\right)$ and $\left(\frac{5}{3}, \infty\right) . “$

A consequence of the answer to example 1 is that we can draw the graph of $f(x)=\frac{2 x-9}{3 x-5}$ in two pieces: one to the left of $\frac{5}{3}$ and one to the right of $\frac{5}{3}$. Figure 5 shows the graph of $f$ as drawn by Mathematica.
Example 2 On what intervals is $f(x)=\frac{\sqrt{x+4}}{x+1}$ continuous?
Solution We need to know (1) where we are dividing by zero and (2) where the square root is defined. For (1), set the denominator equal to zero and solve, yielding $x=-1$, which must be excluded from the domain. For (2), we set the expression under the square root to be at least zero and solve: \begin{aligned} x+4 & \geq 0 \ x & \geq-4 . \end{aligned}
Figure 6 contains number lines picturing solutions to (1) and (2) and their intersection; $f$ is continuous on the intervals $[-4,-1)$ and $(-1, \infty)$

A consequence of the answer to example 2 is that we can draw the graph of $f(x)=\frac{\sqrt{x+4}}{x+1}$ in two pieces: one between $-4$ and $-1$ (including $-4$ but not $-1$ ), and one to the right of $-1$.

# 微积分代考

## 数学代写|微积分代写Calculus代写|Continuity of polynomial and rational functions

$$f(k)=b k^2+c k+d$$

$$\lim x \rightarrow k\left(b x^2+c x+d\right)=b(k+\alpha)^2+c(k+\alpha)+d \quad=b\left(k^2+2 k \alpha+\alpha^2\right)+c k+c \alpha+d=b k^2+2 b k \alpha+b \alpha^2+c k+c \alpha+d \quad \approx b k^2+c k$$

## 数学代写|微积分代写Calculus代写|Where is f continuous?

$$3 x-5=03 x \quad=5 x=\frac{5}{3} .$$

$$x+4 \geq 0 x \quad \geq-4 .$$

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## MATLAB代写

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