LIMITS & CONTINUITY (1.7 & 1.8) NAME____________________________ 1. Find each limit. Include a table of values to illustrate your answer. Include two tables if you need to consider a two sided limit.

A. 1

0lim(1 ) x

xx

→+ = B.

0

sin(2 )limθ

θθ→

=

C. 2 2

lim5 6y

yy→∞

+=

− D.

1

1lim

1t

tt+→

−=

−

2. Find each of these limits. Use the limits to sketch a graph. Be sure to include any asymptotes, holes, or other important characteristics.

2( )2

xf xx−

=−

y

x6

6

-6

-6

lim ( )x

f x→−∞

= lim ( )x

f x→∞

=

2lim ( )

xf x

−→−=

2lim ( )

xf x

+→−=

2lim ( )x

f x→

=

3. Find each of these limits. Use the limits to sketch a graph. Be sure to include any asymptotes, holes, or other important characteristics.

θ

( ) ln sing θ θ= lim ( )

ng

θ πθ

+→= For 0, 1, 2, 3, n = ± ± ±

lim ( )

ng

θ πθ

−→= For 0, 1, 2, 3, n = ± ± ±

4. Find each of these limits. Use the limits to sketch a graph. Be sure to include any asymptotes, holes, or other important characteristics.

( ) cos(2 )rh r e r−= lim ( )

rh r

→−∞= lim ( )

rh r

→∞=

5. Find the value of k that would make the function continuous in each case.

A. 1 0( )

0

xe xg x xk x

⎧ −≠⎪= ⎨

⎪ =⎩

B.

sin(5 ) 1 12 1 2( )

12

x xxh xk x

π −⎧ ≠⎪⎪ −= ⎨⎪ =⎪⎩

6. Find the value of k that would make the limit exist. Find the limit.

A. 32 6lim

3kx

xx→∞

−+

B. 2

2

10lim2x

x kxx→

+ −−

7. In each case sketch a graph with the given characteristics. A. (4)f is undefined and

4lim ( ) 2x

f x→

=

B. and (3) 2f =

3lim ( )x

f x→

does not exist.

C. and (1) 3f =

1lim ( ) 2x

f x→

= −