i need some serious help with limits in pre-calc.

here are a few questions that i really do not understand.

1. Evaluate: lim (3x^3-2x^2+5) x--> -1

2. Evaluate: lim [ln(4x+1) x-->2

3. Evaluate: lim[cos(pi x/3)] x-->2

4. Evaluate: lim x^2+x-6/x^2-9 x--> -3

5. Evaluate: lim x^2-5x+4/x^2-2x-8 x->4

6. For the function x^2+2x+4 find lim f(h+4)-f(4)/h h--> 0

Seizing the means of production

Sure, I can help you with those questions on limits.

1. To evaluate the limit lim (3x^3-2x^2+5) as x approaches -1, you can simply substitute -1 into the expression. So, replace every x with -1 in the given expression:

lim (3(-1)^3-2(-1)^2+5)

Simplifying this expression gives us:

lim (3(-1)+2+5) = lim (-3+2+5) = lim (4) = 4

Therefore, the limit is 4.

2. To evaluate the limit lim [ln(4x+1)] as x approaches 2, similarly substitute 2 into the expression:

lim [ln(4(2)+1)]

Simplifying this expression gives us:

lim [ln(9)]

The natural logarithm of 9 is approximately 2.1972.

Therefore, the limit is approximately 2.1972.

3. To evaluate the limit lim[cos(pi x/3)] as x approaches 2, substitute 2 into the expression:

lim[cos(pi*2/3)]

The cosine of 4pi/3 is -0.5.

Therefore, the limit is -0.5.

4. To evaluate the limit lim [(x^2+x-6)/(x^2-9)] as x approaches -3, substitute -3 into the expression:

lim [((-3)^2+(-3)-6)/((-3)^2-9)]

Simplifying this expression gives us:

lim [(9-3-6)/(9-9)] = lim [0/0]

This expression gives us an indeterminate form 0/0, so we need to simplify further. Factoring the numerator and denominator:

lim [(x-2)(x+3)/(x-3)(x+3)]

Canceling out the common factors:

lim [(x-2)/(x-3)]

Substituting -3 into the expression:

lim [(-3-2)/(-3-3)]

This simplifies to:

lim [-5/-6] = 5/6

Therefore, the limit is 5/6.

5. To evaluate the limit lim [(x^2-5x+4)/(x^2-2x-8)] as x approaches 4, substitute 4 into the expression:

lim [(4^2-5(4)+4)/(4^2-2(4)-8)]

Simplifying this expression gives us:

lim [(16-20+4)/(16-8-8)] = lim [0/0]

This expression gives us an indeterminate form 0/0, so we need to simplify further. Factoring the numerator and denominator:

lim [(x-4)(x-1)/(x-4)(x+2)]

Canceling out the common factors:

lim [(x-1)/(x+2)]

Substituting 4 into the expression:

lim [(4-1)/(4+2)]

This simplifies to:

lim [3/6] = 1/2

Therefore, the limit is 1/2.

6. To find lim [f(h+4)-f(4)/h] as h approaches 0, we need to substitute h=0 into the expression:

lim [( (h+4)^2+2(h+4)+4 - (4^2+2(4)+4) ) / h]

Simplifying this expression gives us:

lim [(h^2+8h+16+2h+8+4-12-8-4)/h]

lim [(h^2+10h+16-12-8-4)/h]

lim [(h^2+10h-12)/h]

Now, we factor the numerator:

lim [(h-2)(h+6)/h]

Canceling out the common factor:

lim [(h+6)]

Substituting h=0 into the expression:

lim [(0+6)] = 6

Therefore, the limit is 6.

I hope these explanations helped you understand how to evaluate limits in pre-calc.

Sure! I'll walk you through each question step-by-step:

1. Evaluate: lim (3x^3-2x^2+5) x--> -1
To evaluate this limit, we substitute -1 for x in the expression:
lim (3x^3-2x^2+5) = 3(-1)^3-2(-1)^2+5 = 3(-1)-2(1)+5 = -3-2+5 = 0.

2. Evaluate: lim [ln(4x+1)] x-->2
To evaluate this limit, we substitute 2 for x in the expression:
lim [ln(4x+1)] = ln(4(2)+1) = ln(8+1) = ln(9).

3. Evaluate: lim[cos(pi x/3)] x-->2
To evaluate this limit, we substitute 2 for x in the expression:
lim[cos(pi x/3)] = cos(pi(2)/3) = cos(2pi/3).

4. Evaluate: lim (x^2+x-6)/(x^2-9) x--> -3
To evaluate this limit, we substitute -3 for x in the expression:
lim (x^2+x-6)/(x^2-9) = ( (-3)^2 + (-3) - 6 ) / ( (-3)^2 - 9 )
= (9 - 3 - 6) / (9 - 9) = 0/0.
Note that we got an indeterminate form 0/0. We can further simplify this using factoring or algebraic manipulation.

5. Evaluate: lim (x^2-5x+4)/(x^2-2x-8) x->4
To evaluate this limit, we substitute 4 for x in the expression:
lim (x^2-5x+4)/(x^2-2x-8) = (4^2 - 5(4) + 4)/(4^2 - 2(4) - 8)
= (16 - 20 + 4)/(16 - 8 - 8) = 0/0.
Again, we got the indeterminate form 0/0.

6. For the function f(x) = x^2+2x+4, find lim [f(h+4)-f(4)]/h h--> 0
To find this limit, we substitute 0 for h in the expression:
lim [f(h+4)-f(4)]/h = [ (h+4)^2 + 2(h+4) + 4 - (4^2 + 2(4) + 4) ] / h
= [ h^2 + 8h + 16 + 2h + 8 - 16 - 8 - 4 ] / h
= ( h^2 + 10h + 4 ) / h.
We cannot simplify this any further without knowing the value of h.

I hope this helps! Let me know if you have any further questions or need more explanations.