Calculus I Assignment # 1

This assignment covers

A: Algebraic and Trigonometric Limits

B: Limits on Restricted Domains

C: Continuity

QUESTIONS

A) Evaluate These Limits

If the limit does not exist, state why.

1) 2) 3)
     
4) 5) 6)
     

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B) Limits on Restricted Domains (piecewise defined functions)

Discuss restrictions on the domains and the limits for these functions.

1) 2)

   
3)  
   

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C) Continuity

1) Discuss the discontinuities of these functions:
For removeable jumps, find the limit.

1) 2)
   
3) 4)
   

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2) Find values for c and/or c and d so that f(x) is continuous for all Real numbers.

1) 2)

   
3) 4)

   

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SOLUTIONS

A) Evaluate These Limits

If the limit does not exist, state why.

1) 2)
   
3) 4)
   
5) 6)
   

.

B) Limits on Restricted Domains (piecewise defined functions)

Discuss restrictions on the domains and the limits for these functions.

1)

so limit = 7

2)

so limit = 1

   
3)

therefore, limit doesn't exist.

 
   

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C) Continuity

1) Discuss the discontinuities of these functions:
For removeable jumps, find the limit.

1) is restricted to -3 [ x [ 3
and x 4 - 16 has vertical asymptotes at x = ± 2,
so the function is continuous on:
-3 [ x < -2 and -2 < x < 2 and 2 < x [ 3.
2) is continuous for x m 0
and x² - 1 has vertical asymptotes at x = ± 1
the function is continuous on
x m 0, x ! 1.
   
3)

continuous for all Real numbers, x ! 2
removeable jump at x = 2, y = 1/12.

4) -2 [ x [ 3 ensures that the expression
is always positive or zero so that the
square root exists.
   

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2) Find values for c and/or c and d so that f(x) is continuous for all Real numbers.

1)

set them equal, solve for c = 5/2 or 2.5

2)

set them equal, solve for c = - 2.

   
3)

Now we solve the system of 2 equations
for c = -2 and d = -6.

4)

(c - 2)(c - 1) = 0 gives solution c = 1 or 2.

   

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