Calculating the Limit of $\displaystyle \frac{5^{\sin{h}}-1}{\tan{h}}$

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In summary, the conversation discusses finding the limit of a given function using L'Hôpital's Rule. The formula is rearranged and simplified through the use of substitution and differentiation to arrive at the final answer of $\log(5)$. It is determined that L'Hôpital's Rule is indeed the best method for solving this problem.
  • #1
Guest2
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With this one I don't know where to start

$\displaystyle \lim_{h \to 0} \frac{5^{\sin{h}}-1}{\tan{h}}$
 
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  • #2
Guest said:
With this one I don't know where to start

$\displaystyle \lim_{h \to 0} \frac{5^{\sin{h}}-1}{\tan{h}}$

I would use L'Hôpital's Rule...:)
 
  • #3
MarkFL said:
I would use L'Hôpital's Rule...:)
Is this correct?

$ \ell :=\displaystyle \lim_{h \to 0} \frac{5^{\sin{h}}-1}{\tan{h}} $ let $t = \tan{h}$ then $ \sin(h) = \sin(\arctan{t}) = \frac{t}{\sqrt{t^2+1}}$.

$$\begin{aligned} \displaystyle \ell & = \lim_{t \to 0} \frac{5^{\dfrac{t}{\sqrt{t^2+1}}}-1}{t} \\& =\lim_{t \to 0} \frac{5^{\dfrac{t+x}{\sqrt{(t+x)^2+1}}}-5^{\dfrac{x}{\sqrt{x^2+1}}}}{t}\bigg|_{x=0} \\& = \frac{d}{dx} 5^{\dfrac{x}{\sqrt{x^2+1}}}\bigg|_{x=0} \\& = \frac{5^{\dfrac{x}{\sqrt{x^2+1}}}\log(5)}{\sqrt{(x^2+1)^3}}\bigg|_{x=0} \\& = \log(5). \end{aligned}$$
 
  • #4
Never mind, L'hopital is indeed the best way to do this.
 

FAQ: Calculating the Limit of $\displaystyle \frac{5^{\sin{h}}-1}{\tan{h}}$

What is the purpose of calculating the limit of this expression?

The limit of this expression can provide insight into the behavior of the function as the variable approaches a certain value. It can also help determine the existence of a horizontal asymptote.

What is the general process for calculating the limit of this expression?

The general process involves substituting the variable with the value it approaches, simplifying the expression, and evaluating the resulting limit.

What is the significance of the trigonometric functions in this expression?

The trigonometric functions, specifically sine and tangent, represent the relationship between angles and sides in a right triangle. In this expression, they are used to manipulate the variable and determine the limit.

How does the exponent affect the limit of this expression?

The exponent, in this case the sine function, affects the limit by changing the rate of change of the function as the variable approaches a certain value. It can also determine the existence of a vertical asymptote.

Are there any special cases to consider when calculating the limit of this expression?

Yes, there are certain values of the variable that may result in an undefined limit, such as when the denominator approaches 0. These cases require further analysis to determine the limit.

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