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)

Verified step by step guidance

1. Identify the type of limit problem: This is a polynomial function, which is continuous everywhere.
2. Recall that for continuous functions, the limit as x approaches a value is simply the function evaluated at that value.
3. Substitute x = 1 into the polynomial: 2(1)^3 - 3(1)^2 + 4(1) + 5.
4. Simplify the expression by performing the arithmetic operations: calculate each term separately.
5. Combine the results of the arithmetic operations to find the limit.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for evaluating continuity and differentiability. In this case, we are interested in the limit of a polynomial function as x approaches 1.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. They are continuous and differentiable everywhere on their domain, making it straightforward to evaluate limits at any point. The function given in the question, 2x^3−3x^2+4x+5, is a polynomial, which simplifies the limit evaluation process.

Substitution Method

The substitution method is a technique used to evaluate limits by directly substituting the value that x approaches into the function, provided the function is continuous at that point. For the limit as x approaches 1, we can substitute 1 into the polynomial to find the limit's value, as polynomials do not have discontinuities.