Question

, find each of the right-hand and left-hand limits or state that they do not exist. $$ \lim _{x \rightarrow 1^{-}} \frac{\sqrt{1+x}}{4+4 x} $$

Short answer

The left-hand limit is \( \frac{\sqrt{2}}{8} \).

Step-by-step solution

Identify the Approach to Solve the Left-Hand Limit

To find the left-hand limit, we are given an expression \( \lim _{x \rightarrow 1^{-}} \frac{\sqrt{1+x}}{4+4 x} \). This means we are interested in what happens to \( \frac{\sqrt{1+x}}{4+4 x} \) as \( x \) approaches 1 from the left.

Evaluate the Numerator

The numerator is \( \sqrt{1+x} \). As \( x \rightarrow 1^{-} \), \( 1+x \rightarrow 2 \). Thus, \( \sqrt{1+x} \rightarrow \sqrt{2} \).

Evaluate the Denominator

The denominator is \( 4 + 4x \). As \( x \rightarrow 1^{-} \), \( 4x \rightarrow 4 \). Therefore, \( 4 + 4x \rightarrow 8 \).

Find the Left-Hand Limit

Substituting the limits of the numerator and denominator from previous steps into the fraction, we find \( \lim _{x \rightarrow 1^{-}} \frac{\sqrt{1+x}}{4+4 x} = \frac{\sqrt{2}}{8} \). This limit exists.

Key concepts

Limits

In calculus, limits are a fundamental concept that describes the behavior of a function as it approaches a certain point. Understanding limits is crucial because it forms the basis for both derivatives and integrals, which are the two main ideas in calculus. Limits help us investigate what happens to a function's value as the input value gets closer and closer to a specified number, and they are essential tools when dealing with functions that have discontinuities or undefined points.

When we write a limit expression, it looks like this: \( \lim_{x \to c} f(x) \), where

• \(x\) is the variable of the function \(f(x)\),
• \(c\) is the point that \(x\) is approaching, and
• \(f(x)\) is the function whose behavior we are analyzing.

By evaluating limits, we can predict and understand the behavior of functions at points where they are not necessarily defined directly, providing a deeper insight into their graphical and numerical properties.

Left-hand limit

A left-hand limit is a type of limit denoted as \( \lim_{x \to c^{-}} f(x) \) and it specifically looks at what happens to the function \( f(x) \) as \( x \) approaches \( c \) from the left-hand side. This is different from a regular limit, which examines the function's behavior from both sides of \( c \).

Understanding left-hand limits is vital for functions that behave differently as they approach a certain point from the left compared to the right. In the given exercise, calculating the left-hand limit for the function \( \frac{\sqrt{1+x}}{4+4x} \) tells us the tendency of the function's value as \( x \) nears 1 from values less than 1.

Such an analysis helps in understanding the continuity and the potential points of discontinuity of the function. In this case, by evaluating \( \lim_{x \to 1^{-}} \frac{\sqrt{1+x}}{4+4x} \), we focus solely on the behavior from values less than 1.

Numerator and denominator evaluation

Evaluating the numerator and denominator separately is a crucial step when dealing with rational functions in limits. It simplifies the understanding of the overall behavior of the expression.

For the given function in the exercise, \( \frac{\sqrt{1+x}}{4+4x} \), we first address:

• The numerator \( \sqrt{1+x} \): As \( x \) approaches 1 from the left, \( 1+x \) tends towards 2, leading the square root to approach \( \sqrt{2} \).
• The denominator \( 4+4x \): Similarly, as \( x \) gets close to 1 from the left, \( 4x \) nears 4, and thus \( 4+4x \) tends towards 8.

After evaluating both parts separately, these results can be combined back to find the behavior of the whole function as \( x \to 1^{-} \). This approach helps in assessing whether limits are finite and exist smoothly.

Approaching limits from the left

Approaching limits from the left implies considering the direction from which \( x \) approaches the target value. This directional approach can show if a function is continuous from a specific side or if discontinuities occur from that direction.

In our exercise case, \( \lim_{x \to 1^{-}} \frac{\sqrt{1+x}}{4+4x} \), the notation \( x \to 1^{-} \) indicates that we are only interested in the values of \( x \) slightly less than 1. By analyzing these values, we gain insight into whether there are abrupt changes in the function as \( x \) nears 1 but remains less than it.

Understanding limits from a specific direction is essential in calculus as it helps in:

• Determining the smooth compatibility of the function at that point,
• Identifying potential jumps that might cause issues in real-world applications, and
• Ensuring accurate interpretations for engineering or physics-related problems.

By interpreting these left-hand approaches, one can ensure detailed and precise understanding of a function’s behavior.