Question

Evaluate the following limits. $$\lim _{x \rightarrow \infty} \frac{\ln (3 x+5)}{\ln (7 x+3)+1}$$

Short answer

Question: Evaluate the limit of the following function as x goes to infinity: \(\lim _{x \rightarrow \infty} \frac{\ln (3 x+5)}{\ln (7 x+3)+1}\) Answer: \(\frac{\ln (3)}{\ln (7)+1}\)

Step-by-step solution

Divide both the numerator and denominator by x

Divide the terms inside the logarithms in the numerator and the denominator by x: $$ \lim _{x \rightarrow \infty} \frac{\ln \left(\frac{3 x+5}{x}\right)}{\ln \left(\frac{7 x+3}{x}\right)+1} $$

Simplify the divided expressions

The expressions inside the logarithms can be simplified by dividing each term by x: $$ \lim _{x \rightarrow \infty} \frac{\ln (3+\frac{5}{x})}{\ln (7+\frac{3}{x})+1} $$

Apply the limit properties

Apply the properties of limits and ln to simplify further: $$ \lim _{x \rightarrow \infty} \frac{\ln (3+\frac{5}{x})}{\ln (7+\frac{3}{x})+1} = \frac{\ln (\lim _{x \rightarrow \infty} (3+\frac{5}{x}))}{\ln (\lim _{x \rightarrow \infty} (7+\frac{3}{x}))+1} $$

Evaluate the limits

Evaluate the limits of the expressions inside the logarithms: $$ \frac{\ln (\lim _{x \rightarrow \infty} (3+\frac{5}{x}))}{\ln (\lim _{x \rightarrow \infty} (7+\frac{3}{x}))+1} = \frac{\ln (3+0)}{\ln (7+0)+1} $$

Simplify the expressions

Simplify the expressions inside and outside the logarithms to get the final answer: $$ \frac{\ln (3+0)}{\ln (7+0)+1} = \frac{\ln (3)}{\ln (7)+1} $$ So, the limit is equal to \(\frac{\ln (3)}{\ln (7)+1}\).

Key concepts

Logarithms

Logarithms are a fundamental concept in mathematics where we solve for the exponent that a base number must be raised to in order to produce a given number. For instance, the expression \(\ln(x)\) refers to the natural logarithm of x, which uses the base e (approximately 2.718).
In calculus and limit problems like the one we are examining, logarithms are utilized to transform multiplication into addition. This property is extremely useful when simplifying complex expressions.

• The inverse relation between logarithms and exponentials helps solve equations involving growth rates and decay.
• Logarithms have specific rules such as the product rule, quotient rule, and power rule which support the simplification process during limit evaluation.

By applying these logarithmic rules, we can decompose and manage the components of the given limit expression to evaluate it systematically.

Infinity Limits

Infinity limits deal with the behavior of functions as the variable approaches infinity. This means we are interested in the value the function seems to get closer to, as x becomes infinitely large. Understanding this concept allows us to predict end-behavior of functions, which is crucial in calculus.
As x approaches infinity in functions such as \(3x + 5\) or \(7x + 3\), the impact of constant terms like 5 and 3 diminishes. That's because these constants become negligible when compared to the scale of infinity. Thus, terms like \(\frac{5}{x}\) or \(\frac{3}{x}\) tend towards zero.
Recognizing that constants disappear in such a way allows us to simplify complex limits and ultimately evaluate the behavior of functions as x grows larger. By focusing on the predominant term, here 3 in \(3x\) and 7 in \(7x\), we simplify the limit to its core components.

Limit Properties

The properties of limits are powerful tools that let us break down complex problems into simpler, solvable parts. These properties include the limit of a constant, the limit of a sum, and the quotient rule for limits.
Applying the properties of limits helps in reformulating tricky expressions. Specifically, in our given problem, the limit properties are applied to simplify the expression \(\ln\left(3+\frac{5}{x}\right)\) as x tends to infinity.

• The Limit of a Sum states that the limit of \(a+b\) is equal to \(a\) plus the limit of \(b\).
• The Limit of a Quotient helps solve expressions where the overall limit may be a fraction of two functions.

These properties enable the breaking down of complex logarithmic forms into simpler parts that can be analyzed individually and then recombined to find the final result of the limit.