Limits and Continuity Practice Problems With Solutions :
Here we are going to see some 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)
(2) lim x->2 (x - 2)/(x2 - 4)
(3) lim x -> 0 (√(x + 3) - √3)/x
(4) lim x->-3 (√(1-x) - 2)/(x + 3)
(5) lim x->0 sin x/x
(6) lim x -> 0 (cos x - 1)/x
Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
(7) lim x->3 (4 - x)
(8) lim x->1 (x2 + 2)
(9) lim x->2 f(x)
Where f(x) = 4 - x x ≠ 2
0 x = 2
(10) lim x->1 f(x)
Where f(x) = x2 + 2 x ≠ 1
= 1 x = 1
(11) lim x->3 1/(x- 3)
(12) lim x->5 |x - 5|/(x - 5)
(13) lim x->1 sin πx
(14) lim x->1 sec x
(15) lim x->π/2 tan x
(16) Sketch the graph of f, then identify the values of x0 for which 0 lim x -> x0 f(x) exists.
(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)
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