LIMITS AND CONTINUITY PRACTICE PROBLEMS WITH SOLUTIONS

Complete the table using calculator and use the result to estimate the limit.

(1)  lim x->2 (x - 2)/(x2 - x - 2)     

Solution

(2)   lim x->2 (x - 2)/(x2 - 4)

Solution

(3)  lim x -> 0 (√(x + 3) - √3)/x

Solution

(4)  lim x->-3 (√(1-x) - 2)/(x + 3)

Solution 

(5)  lim x->0 sin x/x 

Solution

(6)   lim x -> 0 (cos x - 1)/x

Solution

Use the graph to find the limits (if it exists). If the limit does not exist, explain why?

(7)  lim x->3 (4 - x)

Solution

(8)  lim x->1 (x2 + 2)

Solution

(9)  lim x->2 f(x)  

Where f(x)  =  4 - x    x ≠ 2

                       0          x = 2

Solution

(10)  lim x->1 f(x) 

Where f(x)  =  x2 + 2      x ≠ 1

                    =    1            x = 1

Solution

(11)  lim x->3 1/(x- 3)

Solution

(12)  lim x->5 |x - 5|/(x - 5)

Solution

(13)  lim x->1  sin πx

Solution

(14)  lim x->1  sec x

Solution

(15)  lim x->π/2 tan x 

Solution

(16)  Sketch the graph of f, then identify the values of x0 for which 0 lim x -> x0 f(x) exists.

(i)

(ii)  

(18)  Sketch the graph of a function f that satisfies the given values :

f(0) is undefined

lim x -> 0 f(x)  =  4

f(2)  =  6

lim x -> 2 f(x)  =  3                Solution

(19)  f(-2)  =  0

f(2)  =  0

lim x -> -2 f(x)  =  0

lim x -> 2 f(x)  does not exists              Solution

(20)  Write a brief description of the meaning of the notation lim x -> 8 f(x)  =  25       Solution

(21)  If f(2) = 4, can you conclude anything about the limit of f(x) as x approaches 2?        Solution

(22)  If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? Explain reasoning.    Solution

(23)  Evaluate : 

lim x->3 (x2 -9)/(x - 3) if it exists by finding f(3+) and f(3-)      Solution

(24)  Verify the existence of lim x -> 1 f(x)  

Solution

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