OSH/Example/2

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Example problem statement

For what value of $a$ is the function $f(x)$ continuous at all points, where $f(x)$ is the following piecewise function $$ f(x) = \left\{ \begin{array}{ll} x^2+ax & \quad x < 1 \\ 4-x & \quad x \geq 1 \end{array} \right. $$


This problem requires the calculation of a limit. Students often abuse the limit notation when writing up such calculations. Do not drop the "$\lim$" in front of the function in the middle of a multi-line calculation and do not abuse the equal sign.

Solution 1

$$\lim_{x\to 1^-}f(x) = \lim_{x\to 1^-}x^2+ax = 1+a.$$ $$\lim_{x\to 1^+}f(x) = \lim_{x\to 1^+}4-x = 3.$$ $$1+a=3$$ $$a=2$$

Comments

Too terse.


Solution 2

The component parts of $f$ are polynomials and we saw in class (and in the course notes - page 458, line 12) that polynomials are continuous. This means that away from $x=1$, this function is continuous. The only point that whose continuity is in question is the point $x=1$. To check continuity at this point, we calculate the limit of $f(x)$ as $x$ approaches 1 from the left and get $$\lim_{x\to 1^-}f(x) = \lim_{x\to 1^-}x^2+ax = 1+a.$$ This last step is justified because as I already mentioned, the function is just a polynomial on either side of $x=1$ and so continuous. That means we can evaluate the left sided limit simply by evaluating the function at the limiting value. Next we calculate the limit of $f(x)$ as $x$ approaches 1 from the right, $$\lim_{x\to 1^+}f(x) = \lim_{x\to 1^+}4-x = 3.$$ Here again, the last step is justified by the fact that the function is a polynomial on either side of $x=1$ and hence continuous. That means we can evaluate the left sided limit simply by evaluating the function at the limiting value. For $f$ to be continuous, we require that $1+a=3$. Solving this for $a$ we find that $a=2.$

Comments

Too much detail. Repetitive writing. A box around the answer - nice touch, easy for the marker to see the important stuff.


Solution 3

As the component parts of $f$ are polynomials and hence continuous on their own, the only potential discontinuity is at $x=1$. Calculating the limit from the left, $$\lim_{x\to 1^-}f(x) = \lim_{x\to 1^-}x^2+ax = 1+a.$$ And from the right, $$\lim_{x\to 1^+}f(x) = \lim_{x\to 1^+}4-x = 3.$$ For $f$ to be continuous, we require that $1+a=3$ or that $a=2.$

Comments

This is the style you should emulate.

  1. The equations are set out on a line by themselves so that the marker has an easy time looking over those without having to read the text. The equations should not be buried in paragraphs.
  2. The text is there to provide explanation if necessary but not with excessive detail or description.
  3. The most important equation(s) is(are) boxed so as to be easily spotted.