Page 214 - DMTH201_Basic Mathematics-1

Unit 7: Limits

2. Find the following limit if it exists. Specify any horizontal or vertical asymptotes of the Notes
graph of the function.

Solution

As x – , cos x keeps ocillating between 1 and –1, so cos x keeps oscillating between
2
2
0 and 1, thus cos x + 1 keeps oscillating between 1 and 2. Consequently,
2
doesn’t exist. There are no horizontal asymptotes. As cos x + 1 is defined everywhere, there

are no vertical asymptotes.
3. Let

Determine:

Specify horizontal and vertical asymptotes if any.
Solution
2
As x , cos x + 1 keeps oscillating between 1 and 2. So:

Also:

Thus doesn’t exist.

The horizontal asymptote is the x­axis. The vertical asymptote is the y­axis.

4. Let

Determine:

Specify horizontal and vertical asymptotes if any.
Solution

When x –7+, we have 2x – 1 –15 < 0 and x + 7 0 and x + 7 > 0, so:

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