Vault Of Euclid

Evaluating a Logarithmic Limit Using the Definition of the Derivative

Logarithmic and exponential functions core

Evaluate \( \lim_{x\to 3} \frac{\ln (1+2x) - \ln 7}{x-3} \).

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What you need first

The limit definition of the derivative
Evaluating limits by algebraic substitution
The derivative of the natural logarithm function

What this problem tests

Tests the ability to recognize the limit definition of a derivative in a disguised form and to apply algebraic substitutions to limits.

This problem is typical of first-semester calculus courses when students are learning to connect limits with derivatives.

More on Logarithmic and exponential functions

Simplifying the Natural Logarithm of a Unit Fraction warmup
Volume of Revolution Yielding a Logarithm warmup
Limit of ln(1+x)/x as x approaches 0 warmup
Uniqueness of Solutions to f'(x) = f(x) core
Integral of x tanh(x^2) warmup
Arc Length of a Catenary Curve core

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