Question

$$\lim _{x \rightarrow 0} \frac{x^{3}-5 x^{2}}{x^{2}}$$

Short answer

Answer: The limit of the function as x approaches 0 is -5.

Step-by-step solution

Factor out the common term x^2 from the numerator

First, let's factor out the common term x^2 from the numerator: $$\frac{x^{3}-5 x^{2}}{x^{2}} = \frac{x^2(x-5)}{x^2}$$

Simplify the rational function

Now, we cancel out the common term x^2 in the numerator and the denominator: $$\frac{x^2(x-5)}{x^2} = x - 5$$

Evaluate the limit

Now that we have simplified the function, we can find the limit as x approaches 0 by substituting x = 0: $$\lim _{x \rightarrow 0} (x - 5) = 0 - 5 = -5$$ So the limit of the given function as x approaches 0 is -5.

Key concepts

Factoring Polynomials

Polynomials can seem complex, but they often contain terms that make them easier to handle through factoring. Factoring a polynomial involves expressing it as a product of its simpler components or factors. In the given exercise, we start with a polynomial in the numerator, specifically \(x^3 - 5x^2\).

The first step toward simplifying the limit expression is to identify common factors in the terms of the polynomial. Here, both terms \(x^3\) and \(5x^2\) share \(x^2\) as a common factor. Factoring \(x^2\) out gives us \(x^2(x - 5)\).

By factoring, we convert a more complicated expression into a simpler one, making subsequent steps like simplifying rational expressions possible. This technique is crucial in not only solving limits but also in solving various algebraic equations and simplifying expressions in calculus.

Simplifying Rational Expressions

Once we've factored the polynomial, the next step is simplifying the rational expression. A rational expression is a fraction where both the numerator and the denominator are polynomials. In the exercise, the fraction \(\frac{x^3 - 5x^2}{x^2}\) can be rewritten using our factored form as \(\frac{x^2(x-5)}{x^2}\).

Simplifying rational expressions involves canceling out the common terms in the numerator and denominator. The \(x^2\) in both the numerator and denominator can be canceled out, resulting in the simpler expression \(x - 5\).

Caution is needed when simplifying, as you must ensure you are not dividing by zero. Simplifying makes it easier to evaluate expressions by reducing them to their simplest form.

Evaluating Limits

Evaluating limits involves finding the value that a function approaches as the input approaches a given point. In our exercise, we are tasked with finding \(\lim_{x \to 0}(x - 5)\). Once the rational expression is simplified to \(x - 5\), it becomes straightforward to evaluate the limit.

To find this limit, we substitute \(x = 0\) into the expression. This straightforward substitution yields \(0 - 5 = -5\). Therefore, the limit of the original expression as \(x\) approaches 0 is -5.

Knowing how to evaluate limits is essential in calculus as they form the basis for derivative calculations and solving real-world problems involving rates of change and motion.