Evaluate lim x→1 (x^3+3x^2−3x+1).

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Identify the type of limit problem: This is a direct substitution problem because the function is a polynomial.

Recall that for polynomial functions, if the function is continuous at the point of interest, you can directly substitute the value of x into the function.

Substitute x = 1 into the polynomial: $x^3 + 3x^2 - 3x + 1$.

Calculate each term separately: $1^3$, $3(1)^2$, $-3(1)$, and $+1$.

Add the results of each term to find the limit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of the function as x approaches 1. Understanding limits is crucial for evaluating functions at points where they may not be explicitly defined or for determining their behavior near those points.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function in the limit, x^3 + 3x^2 - 3x + 1, is a polynomial. Evaluating limits of polynomial functions is often straightforward, as they are continuous everywhere, allowing us to directly substitute the value of x into the function.

Continuity

Continuity refers to a property of functions where they do not have any breaks, jumps, or holes in their graphs. A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. Since the polynomial in the limit is continuous at x = 1, we can evaluate the limit by simply substituting x = 1 into the function.

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Textbook Question

Determine $$\lim$_{x$\rightarrow\]\infty$}f$\left$(x$\right$)$ and $$\lim$_{x$\rightarrow$-$\infty$}f$\left$(x$\right$)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).