Mathematics 150AL

Assignment 2 -- Limits and Continuity

Start with a piecewise-defined function:

MATH


s3a2__2.png

The graph appears to be continuous. We know that $\sin x$ and $\arctan x$ are continuous. Thus, in order to show that the piecewise-defined function is continuous, we must check for continuity at $x=0$. That is, we must see whether the function's limit exists at $x=0$ and, if it does, we must see whether it is equal to the function value, $f(0)$ at that point. To see whether the limit exists, we compute the one-sided limits MATH and MATH and check whether they are equal. Since the two limits exist and are equal, the limit itself exists. Since this limit is equal to $f(0)=0, $the function is therefore continuous at $x=0$.

Problems:

For each of the following functions, plot the graph, and check for continuity by computing the one-sided limits.

  1. MATH

  2. MATH

  3. MATH

  4. MATH

  5. MATH

  6. MATH

  7. The function $h(x)$ seems to behave differently than all of the other functions. Describe this behavior in your own words. Most discontinuous functions have jumps. Does $h(x)$ have a jump? Is $h(x)$ continuous?

  8. The function MATH is very useful in mathematics. Evaluate $q(x)$ for $x=$ $1.5,$ $2,$ $2.1,$ $2.3,$ $3,$ $3.5,$ $3.9$ and then sketch the graph of this function. Where is this function not continuous? In your own words, explain what the function does to any given number.

  9. For $p(x)=xq(x)$, sketch the graph and check for continuity.

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