Math10.com

Limits (Intuitive Introduction)

      THE TANGENT PROBLEM
Given a function f(x) and a point P(x₀, y₀) on its graph, find an equation of the line tangent to the graph at P

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      THE AREA PROBLEM
Given a function f, find the area between the graph of the f and an interval [a,b] on the x-axis

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A line is called tangent to a circle if it meets the circle at precisely one point.

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But this definition is not satisfactory for other kind of curves, like

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The line is tangent yet it meets the curve more than once.

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The line occupying this limiting position we consider to be tangent line at P.

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The areas of some plane regions can be calculated by subdividing them into finite number of rectangles or triangle, then adding the area of the constituent parts.

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For many regions a more general approach is needed.

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We approximate the area of this region by inscribing rectangles of equal width under the curve and adding the areas of these rectangles.

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Our approximations will "approach" the exact area under the curve as a "limiting value".

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Let us take
      f(x) = sin(x)/x
Where x is in radians.

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Right hand limit
lim_{x → 0+}   sin(x)/x "The limits of f(x) as x approaches 0 from the right"

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Left hand limit
lim_{x → 0+}   sin(x)/x
"The limit of f(x) as x approaches 0 from left."

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Using a calculator set to the radian mode, we have

x f(x) = sin(x)/x 1.0 0.84107 0.8 0.89670 0.6 0.94107 0.4 0.97355 0.2 0.99335 0.01 0.99998

x f(x) = sin(x)/x -1.0 0.84107 -0.8 0.89670 -0.6 0.94107 -0.4 0.97355 -0.2 0.99335 -0.01 0.99998

As x approach 0 from the left or right, f(x) approach 1.

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The preceding ideas are summarized in this table.

NOTATION	HOW TO READ THE NOTATION
limx→ x0+ = L1	The limit of f(x) as x approaches x0 from the right is equal to L1
limx→ x0- = L2	The limit of f(x) as x approaches x0 from the left is equal to L2
limx→ x0 = L	The limit of f(x) as x approaches x0 is equal to L

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      Numerical Pitfalls
        f(x) = sin(φ/x)
lim_{x → 0} sin(φ/x) = lim_{x → 0+} sin(φ/x) = lim_{x → 0-} sin(φ/x)

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          Table 2.4.4

x	f(x)	x	f(x)
1	0	-1	0
0.1	0	-0.1	0
0.01	0	-0.01	0
0.001	0	-0.001	0
0.0001	0	-0.0001	0

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      Existence of limits

lim_{x→ x0-} f(x) = lim_{x→ x0+} f(x) = lim_{x→ x0} f(x) = +∞
x approaches x₀ from left

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lim_{x→ x0-} f(x) = lim_{x→ x0+} f(x) = lim_{x→ x0} f(x) = -∞
x approaches x₀ from right

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      Example

lim_{x→ x0-} f(x) = -∞       lim_{x→ x0+} f(x) = +∞
The limit of the function does not exist at X₀

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      Example

Here
lim_{x→ -∞} f(x) = -1       lim_{x → +∞} f(x) = 4
So, limit of function does not exist.

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      Example

Here
as x → -∞, f(x) → +∞
as x → +∞, f(x) → -2
So, limit of function does not exist.

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      Example

Here
as x → -∞, f(x) → -∞
as x → +∞, f(x) → oscillates
So, limit of function does not exist.