Page 210 - DMTH201_Basic Mathematics-1

Unit 7: Limits

Notes

Give that the slope of the line is and is passes through the point (2, 3), we can find the equation
of the tangent line using the point­slope formula.
y – y = m(x – m )
1 1
y – 3 =

3y – 9 = x – 2
x – 3y + 7 = 0
The equation of the tangent line to the curve at the point (2, 3) is
x – 3y + 7 = 0.

7.4 The Pinching or Sandwich Theorem

As a motivation let us consider the function

f(x) =

When x get closer to 0, the function fails to have a limit. So we are not able to use the

basic properties discussed in the previous pages. But we know that this function is
bounded below by –1 and above by 1, i.e.

2
for any real number x. Since x 0, we get

Hence when x get closer to 0, x2 and –x2 become very small in magnitude. Therefore any number
in between will also be very small in magnitude. In other words, we have

This is an example for the following general result:
Theorem: The “Pinching” or “Sandwich” Theorem
Assume that
h(x) f(x) g(x)

for any x in an interval around the point a. If

then

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