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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