Question

When the Limit Is an Expression $$\lim _{d \rightarrow 0} \frac{\left[(x+d)^{2}+1\right]-\left(x^{2}+1\right)}{d}$$

Short answer

The limit as \( d \) approaches 0 of the given function is \( 2x \)

Step-by-step solution

Expand the numerator

First, expand the bracket \( (x+d)^2 \) in the numerator to get \( x^2 + 2xd + d^2 + 1 \). Then rewrite the limit expression with the expanded form.

Simplify the expression

Cancel out the \( x^2 + 1 \) terms as they appear in both the subtracted expressions in the numerator. The resulting expression in the numerator is \( 2xd + d^2 \) and the limit becomes \( \lim_{d \rightarrow 0} \frac{2xd + d^2}{d} \).

Factor out d from the numerator

Factor \( d \) out of the numerator to simplify the expression to \( d(2x + d) \). The limit can now be rewritten as \( \lim_{d \rightarrow 0} \frac{d(2x + d)}{d} \).

Cancel the common factor d

Since \( d \) is a common factor in the numerator and the denominator and \( d eq 0 \) as we consider the limit approaching 0, we can cancel out the \( d \) which results in \( \lim_{d \rightarrow 0} (2x + d) \).

Evaluate the limit

Now, take the limit of the expression \( 2x + d \) as \( d \) approaches 0. This results in \( 2x + 0 \) or simply \( 2x \). Therefore, the limit is \( 2x \).

Key concepts

Limit Evaluation

Understanding how to evaluate limits is a fundamental aspect of calculus. When given a limit like \( \lim _{d \rightarrow 0} \frac{\left[(x+d)^{2}+1\right]-\left(x^{2}+1\right)}{d} \), it's crucial to assess the behavior of the function as the variable approaches a certain value—in this case, as \( d \) approaches 0.

To begin with, examine the form of the function. Is it directly substitutable, or does it result in an indeterminate form such as 0/0? If direct substitution isn't possible, algebraic manipulation, like expanding polynomials or factoring, is the next step. Once you simplify the expression, substitution often becomes possible. If the limit exists, it can now be accurately evaluated.

Factoring Algebraic Expressions

Factoring is a powerful algebraic tool that breaks down an expression into simpler components or 'factors' that, when multiplied together, give back the original expression. In the realm of limits and particularly for the problem at hand, factoring allows us to reduce a complex expression and eliminate common factors.

In our workout, factoring is the technique that simplifies \( 2xd + d^2 \) as \( d(2x + d) \). This crucial step reveals a common factor of \( d \) in both the numerator and denominator, setting us up to simplify the fraction further. Without factoring, we'd be stuck with an expression that appears more complex than it actually is.

Simplifying Expressions

Simplifying expressions is the process of reducing them to the most basic form while preserving their value. It's a vital skill in physics, engineering, calculus, and more. This can involve canceling out common factors, combining like terms, or using arithmetic to simplify the terms.

In the given solution, after factoring, we simplify by canceling the common \( d \) term. It's important to note that simplification applies to expressions, not to values resulting from the substitution, as cancelling terms might not be valid if the term is zero. Simplifying makes it easier to evaluate limits, especially when you can't just substitute the variable's approaching value directly.

Basic Calculus

Calculus is all about change and motion and the mathematics that deals with these concepts. Basic calculus includes understanding limits, derivatives, and integrals. Limits, like the one in our problem, help us understand the behavior of functions as variables approach specific values.

Calculating derivatives, for instance, requires the concept of limits to find the instantaneous rate of change, while integrals involve finding areas under curves, which is a form of accumulation. Both of these fundamental operations rely on the concept of limits, demonstrating how foundational the concept of limit evaluation really is in the study of calculus.