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.