Question

, find each of the right-hand and left-hand limits or state that they do not exist. $$ \lim _{x \rightarrow-3^{+}} \frac{\sqrt{3+x}}{x} $$

Short answer

The right-hand limit as \( x \) approaches \(-3\) from the positive side is 0.

Step-by-step solution

Substitute Value into the Function

We need to find the right-hand limit as \( x \) approaches \(-3\) from the positive side for the given function \( \frac{\sqrt{3+x}}{x} \). Start by substituting \( x = -3 \) into the function as a check step. However, \( \sqrt{3+(-3)} = \sqrt{0} = 0 \) and the denominator becomes zero, so we can't directly substitute \( x = -3 \). This suggests a further investigation is needed.

Consider the Behavior of the Numerator

Since we are approaching \( x = -3 \) from the right, the expression \( 3 + x \) is slightly more than zero, making \( \sqrt{3+x} \) a small positive number close to zero. As \( x \rightarrow -3^{+} \), \( \sqrt{3+x} \to 0^+ \).

Consider the Behavior of the Denominator

For \( x \rightarrow -3^{+} \), the values of \( x \) are greater than \(-3\) but very close to it, meaning \( x \) is a small negative number. Thus, as \( x \rightarrow -3^+ \), the denominator \( x \to -3^- \), approximately having values just slightly greater than \(-3\).

Evaluate the Limit Expression's Sign

In this step, consider the fraction \( \frac{0^+}{-3^-} \). Since the numerator approaches zero from the positive side and the denominator approaches \(-3\) from the negative side, the fraction itself tends to a positive result as \(-\frac{0^+}{3^+} = 0^+\).

Conclude the Limit

Since both the numerator and the denominator behavior suggests that the fraction approaches zero from the positive side, conclude that \( \lim_{x \rightarrow -3^{+}} \frac{\sqrt{3+x}}{x} = 0 \).

Key concepts

Understanding Right-Hand Limits

When we talk about a "right-hand limit," we're discussing what happens to a function as the input value, or "x," approaches a certain number from the right or positive side. Imagine you're walking on a number line towards eeding to find out how a function behaves just before you reach that point. In our exercise, we're interested in what happens near the point -3, but just coming from values slightly larger than -3, hence "-3+". This exploration is essential because functions can behave differently as you approach from different sides. In practical terms, while calculating a right-hand limit, you slightly increase the value of the number you're approaching. By checking these values, you gather insights into the behavior of the function near that point. You notice what's happening with the numerator and the denominator individually, forming the overall limit behaviour.

Exploring Left-Hand Limits

Contrary to the right-hand limit, the left-hand limit examines what happens as you approach a certain value from the left or negative side on the number line. This involves gradually getting closer to a point using smaller values. This difference in direction can lead to distinct outcomes, particularly when a graph has sudden changes, dips, or spikes. Think of it as spying on a function from the opposite side as you steadily approach a value like -3 from values less than -3 or "-3-". Doing so helps gather the overall scope of a function's behavior around points where it might change dramatically. In case you were analyzing the left-hand limit in similar scenarios as our exercise, you may notice drastic differences in behavior if the behavior is not uniform.

Numerator and Denominator Behavior: Near Zero

Understanding the behavior of a function as you approach certain limits, like -3 from the right, involves focusing on the numerator and denominator separately. In the given function, \( \frac{\sqrt{3+x}}{x} \), it is crucial to note how both components behave as \( x \to -3^{+} \). * **Behavior of the Numerator**: \( \sqrt{3+x} \) approaches zero from the positive side \(0^+\). Since the square root of any non-negative number remains non-negative, you're watching it creep towards minimal values as \( x \) gets closer to -3 from the right.* **Behavior of the Denominator**: At the same time, for values of \( x \) nearing -3 from the right side, \( x \) itself is just below -3, making it minuscule and slightly negative. When you combine these clues, the numerator approaches zero while the denominator remains negative but close to zero, suggesting the fraction \( \frac{0^+}{-3^-} \) is positively arising, collapsing to zero. Understanding these behaviors paints the full picture of how the given limit results in a positive value close to zero as per the calculations.