Download Sample Questions for Exam 1 (Limits – Sections 2.1 to 2.5) 1. Sketch

Sample Questions for Exam 1
(Limits – Sections 2.1 to 2.5)
1.
Sketch in the grid provided below the graph of a function f that satisfies all of the
following conditions:
lim f ( x)   ,
lim f ( x)  0 ,
lim f ( x)  1 ,
lim f ( x)   ,
x 
x 3
2.
x 
lim f ( x)   ,
x 3
x 3
For the following functions, locate any vertical asymptotes. Justify your answers using
limits.
a)
f ( )  tan   1
b)
g ( x) 
c)
h( x)  1  ln(5  x)
x2
 x  1  x  3
2
3.
Evaluate the following limits:
a)
lim
x  6
b)
x 6
x 6
x 2  2 x 8
lim 4
x 16
x2
c)
t 2 4
lim 3
t 8
t 2
Difference of Cubes Identity: a 3  b3   a  b   a 2  ab  b 2 
d)
lim
x 
e)
3
h
ln(sin x)
lim

x
g)

x 2 1  2 x 3
3
h 1 1

lim
h 0
f)

3x5  2 x3 1

sin 2 x
lim 2
x
x 
4.
Show that the limit given below does not exist.
5.
Let f be the function defined everywhere (i.e. for all real numbers) by
f ( x)   x    x  ,
where   is the floor function defined as follows for any real number x :
 x  = the greatest integer that is less than or equal to x .
Investigate the limit of f as x approaches 1. Does the limit exist? If so, justify your answer.