**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)/(x^{2} - x - 2)

(2) lim _{x->2} (x - 2)/(x^{2} - 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} (x^{2} + 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) = x^{2} + 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 x_{0} for which 0 lim x -> x_{0} 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 (x^{2} -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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