calc exam 1 - limits

Evaluate lim x->infinity sqrt(4x^4+3)/(x^2-4x+2)

2

For f(x)= x^2 + 3x, find f'(2) using the limit definition of the derivative.

7

Which of the following is false?

If f is discontinuous, then cf is discontinuous where c is any real number

Which of the following is the definition of the derivative, f'(a), of f(x) at x=a?

f'(a)= lim h->0 (f(a+h)-f(a))/ h

Consider the function f(x)= (x^2)/(2^(x)+1) on the interval [0,4]. Which of the following equations is NOT guaranteed to have a solution in the interval [0,4] by the Intermediate Value Theorem?

f(x)=1

Which of the following types could potentially be discontinuous on the interval (-infinity, infinity)? (i) Polynomials (ii)Trigonometric (iii)Exponential

ii

Given that lim x->1 g(z)=2, lim x->1 f(z)=4, lim x->1 h(z)=infinity and lim x->1 j(z)=0, which of the following limits is guaranteed both to exist and equal to a finite number?

lim x->1 g(x)/(f(x)-h(x))

Suppose that g(x) is a function and that lim x->9 sqrt(g(x/3) +16)=6. Which of the following is true?

lim x->3 g(x)=20

If f(x)=9/x, then f'(3)=-1; the equation of the tangent line to the graph of f(x) at x=3 is given by:

y= -x+6

Assume that lim x-> infinity f(x)= L. Which of the following statements are true?

y=L is a horizontal asymptote

A particle which moves along a straight line has a position function given by s(t)= 1/(3+t), 0≤ t< infinity, where s(t) is given in meters and t is given in seconds. What is the average velocity in meters per second of the particle on the interval [3,9]?

-1/72

What is the value of the following : lim x-> infinity (x^2-8x+7)/(2x+3x^2+2)?

1/3

Which value of k makes the following function continous? f(x)={ sin(kx) x≤0 { kx^2 x> 0

both 0 and 1

State the equations of all vertical asymptotes for a function f(x) satisfying the following conditions: f(0)=0, lim x->infinity f(x)= infinity, lim x-> -infinty f(x)=-1, lim x-> 4- f(x)= - infinity, lim x-> 4+ f(x)= infinity, lim x->-2 f(x)= infinity

x=4 and x=-2