
ALGEBRA
 solved problems 






The
limit of a function

Limits of
functions
properties 
Assume that 

exist
and that c
is any constant. Then, 




11.
Squeeze rule 
If f
(x)
< g (x) < h (x)
for all x
in an open interval that contains a,
except possibly at x
= a, 
and 



then 




12.
Composition rule 
If f
(x)
is continuous at x
= b
and 



then, 




Limits of
functions
properties use 
Limits
that are commonly used are written below for easy reference, 


By
comparing the area of the sector of the arc x
with areas
of the two right triangles, the smaller triangle
with
legs sin x
and cos x,
and the bigger triangle with legs
tan x
and 1, shown on the right figure, we get 

Since
the left and the right side of the last
inequality tend
to the same limit 1, as x
tends to 0 then, applying
the squeeze rule obtained is 




the
fundamental or basic trigonometric limit.
Note
that the arc length x
is measured in radians. 
Further, 

since both
factors on the right tend to 1, as x
tends to 0, then 











Solved
problems contents  A 



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