Find the following limits or state that they do not exist. Ass...

Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→1(2x^3−3x^2+4x+5)

Accepted Answer

Hello there, today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem, find the limit of the polynomial function F of X equals X to the power of three minus two, multiplied by X to the power of two plus three X minus four as X approaches zero. Awesome. So it appears for this particular problem we're trying to figure out what the limit is of this polynomial function. So now that we know that we're trying to solve for the limit for this particular function, let's read off our multiple choice answers to see what our final answer might be. A is negative four, B is negative two, C is negative 10 and D is eight. Awesome. So first off, in order to solve this problem, we need to identify the polynomial function, which we're told that the polynomial function is F of X is equal to X to the power of three minus two, multiplied by X to the power of two plus which I put a multiplication sign between the two and the X squared, but you don't have to. So it's two multiplied by X to the power two plus three X minus four. Awesome. And the target variable, which we need to identify this, our target variable that we're trying to solve for is the limit as X approaches zero of F of X. So that's our target. I'll put little red stars around it because this is our target, what we're ultimately trying to solve for. So in order to solve this particular problem, let us try the substitution method or more specifically the direct substitution method. So let us substitute in X equals zero into our polynomial function. So we're gonna plug in X so X equals zero. So we're gonna say F of zero is equal to zero to the power of three minus two, multiplied by zero to the power two plus three, multiplied by zero minus four. So when we perform that calculation, we will find that our final answer has to be negative four and that's it we've solved for this problem. Hooray. So that will mean that when we simplify the expression once again, the quickly recap that F of zero is equal to negative four. So therefore, the limit, which once again, the limit as X approaches zero of F of X, which in this case F of X and let we could just write this out. So the limit as X approaches zero of X to the power three minus two multiplied by X squared plus three, X minus four. This will mean that it is equal to negative four, which once again is our final answer. Hooray, we did it. So let me look at our multiple choice answers. The correct answer without a doubt has to be multiple choice answer letter A which once again is negative four. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→1(2x^3−3x^2+4x+5)

Key Concepts

Limits

Polynomial Functions

Substitution Method

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