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Calculus Limits and Continuity

This document discusses limits, continuity of functions, and asymptotes of functions. It contains examples of evaluating limits as variables approach certain values, determining limits using continuity, finding asymptotes, checking continuity of piecewise defined functions, extending functions to make them continuous, and conditions for continuity of piecewise defined functions.

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26 views1 page

Calculus Limits and Continuity

This document discusses limits, continuity of functions, and asymptotes of functions. It contains examples of evaluating limits as variables approach certain values, determining limits using continuity, finding asymptotes, checking continuity of piecewise defined functions, extending functions to make them continuous, and conditions for continuity of piecewise defined functions.

Uploaded by

shuhratjonovbek
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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V. Limits an continuity of functions.

1. Evaluate limits:
x2 −9 x3 +8 2 −1 x3 −2x2 −4x+8
(a) limx→3 3 , (b) limx→−2 x2 −x−6
, (c) limx→1 xx5 −1 , (d) limx→2 x4 −8x2 +16
,
√ −27
x √ √3
2 +1−1
(e) limx→7 x−7 ,
x−6−1
(f) limx→0 1−√x2 +x+1 ,
x
(g) limx→0 √x2 +16−4 ,
x
(h) limx→0 x+1−1
x
,
tg x
(i) limx→π 1+cos
sin2 x
x
, (j) limx→0 2x−sin
x+sin x
x
, (k) limx→ π4 sin 2x
,
sin 5x sin 4x tg x
(l) limx→0 x , (m) limx→0 sin 5x , (n) limx→0 sin 2x ,
x4 +2x−1
(o) limx→0 arctg3x
2x
, (p) limx→−∞ −x3 +4x−1 , (q) limx→−∞ x3 −2x+4 x+2
.

2. Determine limits using continuity of functions at some point:


sin x
 
1−x tg x
(a) limx→0 2 x , (b) limx→+∞ arcsin 1+x , (c) limx→0 arctg x
,
(d) limx→4 log √x−4
x−2
, (e) limx→0 3x·ctgx
.

3. Determine asymptotes of functions given by:


√ √
(a) f (x) = 1−x
x
, (b) f (x) = x − 2 x, (c) f (x) = x2 − 3x + 2,
x3 +8 sin x − 12 x2 +2x
(d) f (x) = x2 −4
, (e) f (x) = x2
, (f) f (x) = e x , (g) f (x) = x+1
.

4. Check if the following functions are continuous:


 
x2 for x < 1, x2 + x − 2 for x < −2,
(a) f (x) = (b) f (x) =
3 − x for x ­ 1, x2 + 5x + 6 for x ­ −2,
 2 
 |x −x−6| for x 6= −2,  cos x−1 for x 6= 0,
(c) if (x) =  x+2
(d) f (x) =  |x|
5 for x = −2, 0 for x = 0.

5. Extend, if it is possible, a given function in such a way that the extended function is continuous on the
whole real line. If:
1
tg 3x tg x−sin x
(a) f (x) = 5x
, (b) g(x) = x3
, (c) h(x) = sin x · sin x1 , (d) p(x) = 2 x−1 .

6. What is a value of p ∈ R which the following function is continuous for? If:



 sin 5x
for x 6= 0,
f (x) =  sin 8x
2p + 1 for x = 0.

7. What are A, B ∈ R which the following function is continuous for? If:



−2 sin x for x ¬ − π2 ,


g(x) = A sin x + B for − π2 < x ¬ π2 ,


cos x for x ­ π2 .

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