Page 207 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes
=

= (as h>0, so|–h|=h)

= –1 ...(iv)

From (iii) and (iv),

Thus, in the first example right hand limit = left hand limit whereas in the second example right
hand limit left hand limit.
Hence the left hand and the right hand limits may not always be equal.
We may conclude that

exists (which is equal to 9) and does not exist.

7.3 Tangents and Limits

A tangent to a curve is a straight line that touches the curve at a single point but does not intersect

it at that point. For example, in the figure to the right, the y­axis would not be considered a
tangent line because it intersects the curve at the origin. A secant to a curve is a straight line that
intersects the curve at two or more points.

In the figure given below, the tangent line intersects the curve at a single point P but does not
intersect the curve at P. The secant line intersects the curve at points P and Q.

The concept of limits begins with the tangent line problem. We want to find the equation of the

tangent line to the curve at the point P. To find this equation, we will need the slope of the tangent

line. But how can we find the slope when we only know one point on the line? The answer is to
look at the slope of the secant line. It’s slope can be determined quite easily since there are two
known points P and Q. As you slide the point Q along the curve, towards the point P, the slope of
the secant line will become closer to the slope of the tangent line. Eventually, the point Q will be
so close to P, that the slopes of the tangent and secant lines will be approximately equal.

A limit of a function is written as:
= L

We want to find the limit of f(x) as x approaches a. To do this, we try to make the values of
f(x) close to the limit L, by taking x values that are close to, but not equal to, a. In short, f(x)
approaches L as x approaches a.

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