Evaluate the following limit: Lim x->0 ((1+x)^4 - 1)/x
Understand the Problem
The question asks to evaluate the limit of the given expression as x approaches 0. We will need to simplify the expression and apply limit rules to find the solution. The expression involves a power function and a fraction, so algebraic manipulation and possibly L'Hôpital's Rule or binomial expansion could be used.
Answer
4
Answer for screen readers
4
Steps to Solve
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Expand $(1+x)^4$ using the binomial theorem The binomial theorem states that $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$. In our case, $a=1$, $b=x$, and $n=4$. Therefore, $$ (1+x)^4 = \binom{4}{0}1^4x^0 + \binom{4}{1}1^3x^1 + \binom{4}{2}1^2x^2 + \binom{4}{3}1^1x^3 + \binom{4}{4}1^0x^4 $$ $$ (1+x)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4 $$
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Substitute the expanded form into the limit expression Now, substitute the expanded form of $(1+x)^4$ into the original limit expression: $$ \lim_{x \to 0} \frac{(1 + 4x + 6x^2 + 4x^3 + x^4) - 1}{x} $$
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Simplify the expression Simplify the numerator: $$ \lim_{x \to 0} \frac{4x + 6x^2 + 4x^3 + x^4}{x} $$
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Factor out x from the numerator Factor out $x$ from the numerator: $$ \lim_{x \to 0} \frac{x(4 + 6x + 4x^2 + x^3)}{x} $$
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Cancel out the x terms Cancel out the $x$ terms in the numerator and denominator: $$ \lim_{x \to 0} (4 + 6x + 4x^2 + x^3) $$
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Evaluate the limit as x approaches 0 Now, evaluate the limit by substituting $x = 0$ into the expression: $$ 4 + 6(0) + 4(0)^2 + (0)^3 = 4 + 0 + 0 + 0 = 4 $$
4
More Information
The limit evaluates to 4. This can also be seen as the derivative of $f(x) = (1+x)^4$ evaluated at $x=0$.
Tips
A common mistake could be to incorrectly expand the binomial expression $(1+x)^4$. Another mistake could be failing to factor and cancel $x$ from the numerator and denominator, leading to an indeterminate form.
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