Expert: Paul Klarreich - 7/2/2007

Question
F(x)=(Sinx)/x when x approaches 0

1. Study Left Hand and Right Hand Limit
2. Conclude your result
3. Sketch the Graph

Answer
Questioner:   Umair
Category:  Calculus
Question:  F(x)=(Sinx)/x when x approaches 0

1. Study Left Hand and Right Hand Limit
2. Conclude your result
3. Sketch the Graph

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Hi, Miss Umair,

Both  sin(x) and  x are 'odd' functions, so
F(x) = F(-x), and the left- and right-hand limits will be the same.

If you try your calculator on values of x near  x = 0, you find that the function values are close to 1.  In fact, on page 137 of your calculus book (it's always on that page) you will find a proof that:
     sin x
lim   ----- = 1
x->0    x

and, of course, the left- and right-hand limits are the same, otherwise the limit would not exist.

Unfortunately I can't make a sketch here, but the graph will have these properties:

A. It is symmetric about the y-axis. (left-right symmetry)
B. It oscillates and crosses the x-axis for all  x = n pi, but the arcs get lower and lower as you get farther from the origin.
C. The point (0,1) is missing.  You draw this little circle about it to show that, like this:


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