Question

Find the limit. $$ \lim _{x \rightarrow 3} \frac{\sqrt{x+1}-1}{x} $$

Short answer

\(\frac{1}{3}\)

Step-by-step solution

Rationalize the numerator

To begin with rationalization, multiply the expression by its conjugate.\[ \lim _{x \rightarrow 3} \frac{\sqrt{x+1}-1}{x} \cdot \frac{\sqrt{x+1}+1}{\sqrt{x+1}+1} \]

Simplify the expression

Simplify the expression by multiplying the numerators and denominators separately. The numerator becomes \(x+1-1\) and the denominator is \(x(\sqrt{x+1}+1)\).\[ \lim _{x \rightarrow 3} \frac{x}{x(\sqrt{x+1}+1)} = \lim _{x \rightarrow 3} \frac{1}{\sqrt{x+1}+1} \]

Substitute x

Now when x is substituted with 3, the expression does not give an indeterminate form.\[ \frac{1}{\sqrt{3+1}+1} = \frac{1}{\sqrt{4}+1} = \frac{1}{2+1} = \frac{1}{3} \]

Key concepts

Rationalizing the Numerator

When faced with a limit problem where the direct substitution leads to an indeterminate form, such as \(0/0\), a common strategy is rationalizing the numerator. This involves multiplying the original expression by a form of 1 that contains the conjugate of the numerator, without changing the value of the expression.

For a numerator with a square root, like \(\sqrt{x+1}-1\), the conjugate would be \(\sqrt{x+1}+1\). This step is critical, as it employs the difference of squares to simplify the square root terms without affecting the limit. By doing so, the troublesome square roots are eliminated, and the resulting expression often resolves the indeterminate form, making it possible to evaluate the limit by direct substitution.

Simplifying Expressions

After rationalizing the numerator, the next step to find the limit of a function is simplifying the expressions. Simplification in the context of limits generally involves expanding products, canceling terms, and combining like terms to create a more straightforward expression.

In the case of rationalized expressions, expanding and canceling out terms ensures that any possible factor in the denominator that could lead to an indeterminate form is removed. This often results in a significant simplification, where the limit may then be directly evaluated. Simplification is not only vital for resolving the specific limit problems, but it's also excellent practice for algebraic manipulation, which is a crucial skill in calculus.

Substitution in Limits

The final phase of evaluating a limit usually involves the technique of substitution. Once we've rationalized the numerator and simplified the expression sufficiently, we substitute the value to which the variable approaches into the simplified function.

If the direct substitution after simplification results in a defined value rather than an indeterminate form, then this value is the limit. If it were still indeterminate, further techniques, such as factoring, L'Hôpital's rule, or perhaps trigonometric identities, would be necessary. Substitution is the simplest and preferred method when applicable, as it quickly yields the desired limit, emphasizing the continuity or discontinuity of the function at that point.