Question

Find the limits, or state that the limit does not exist (you must justify answer):

a) \lim_(x->\infty )(\sqrt(x^(2)+2x)-\sqrt(x^(2)-x))

b) \lim_(x->\infty )(lnx-ln(sin x))

c) \lim_(x->\infty )x^((1)/((lnx)))

Answer #1

Calculate each limit below, if it exists. If a limit does not
exist, explain why. Show all work.
\lim _{x\to 3}\left(\frac{x-3}{\sqrt{2x+3}-\sqrt{3x}}\right)
\lim _{x\to -\infty }\left(\frac{\sqrt{x^2+3x}}{3x+1}\right)

Find the limit if it exists or show that the limit does not
exist.
lim (x,y) to (0,0). y^2 sin^2x/ x^4+y^4

Evaluate the limit using the appropriate properties of limits.
(If an answer does not exist, enter DNE.)
lim x→∞ (6x2 − 7) / (7x2 + x − 3)

Determine each limit or explain why the limit does not
exist.
(a) lim(x,y)→(5,4) : (x − y − 1)/(√(x − y) − 1)
(b) lim(x,y)→(0,1) : (sin(x^(2) + y − 1))/(5y + x^(2) − 5)
(c) lim(x,y)→(3,2) : (4x + y)/(2x − 6)

Find the limits, if they exist, or type DNE for any
which do not exist.
lim(x,y)→(0,0)5x^2/2x^2+y^2
1) Along the xx-axis:
2) Along the yy-axis:
3) Along the line y=mxy=mx :
4) The limit is:

1.
Find the area of the shaded region.
2. Find the limit. Use l'Hospital's Rule where appropriate. If
there is a more elementary method, consider using it.
lim x → 0
1 + 10x
−
1 − 3x
x
3. Find the limit. Use l'Hospital's Rule if appropriate. If
there is a more elementary method, consider using it.
lim x→1
1 − x +
ln(x)
1 +
cos(9πx)
4. Find the limit. Use l'Hospital's Rule if appropriate. If
there is...

Evaluate the limit. (If an answer does not exist, enter
DNE.)
lim x→0+ (9x)sin(4x)

Find the limit, if it exists. (If an answer does not exist,
enter DNE.)
lim (x, y)→(0, 0)
x2 + y2/square root (x2 +
y2+ 25)-5

13. Consider
f(x)=sqrt x-2
a) Using any of the three limit formulas to find f ′ ( a
), what is the slope of the tangent line to f ( x )at x = 18? (6
points execution, 2 points notation)
b) Find the equation of the tangent line at x =
18
14. State the derivative.
a) d/ d x [ x ^n ]
b) d /d x [ cos x ]
c) d /d x [ csc...

1. Evaluate the limit
lim x→0 sin4x /7x
2. In which limits below can we use L'Hospital's Rule? select
all that apply.
lim x→∞ e^−x /x
lim x→0 1−e^x /sin(2x)
lim x→0 x+tanx /sinx
lim x→−∞ e^−x /x

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