1.4 Limits Of Exponential Functionsap Calculus



Answer

(a) Logarithmic (b) Root function (c) Rational function (d) Second-degree polynomial (e) Exponential function (f) Trigonometric function

Work Step by Step

(a) $f(x)=log_{2}x$ contains the base 2 logarithm operator, and thus is a logarithmic function. (b) $g(x)=sqrt[4] x$ is the fourth root of x, and thus is a root function. (c) $h(x)=frac{2x^3}{1-x^2}$ is the quotient of two polynomial functions, and thus is a rational function. (d) $u(x)=1-1.1t+2.54t^2$ has powers of $t$ multiplied by constants. The greatest power of $t$ present is 2, so the function is a second-degree polynomial. (e) $v(t)=5^t$ is the constant $5$ raised to the power of $t$, making this an exponential function. (f) $w(theta)=sin theta hspace{1.5mm} cos^2 theta$ contains the sine and cosine function, both trigonometric functions themselves, making $w(theta)$ a trigonometric function.

Evaluate: $$displaystyle lim_{xto0}, frac x {e^{10x} - 1}$$

Step 1

May 29, 2018 These will all be very useful properties to recall at times as we move throughout this course (and later Calculus courses for that matter). There is a very important exponential function that arises naturally in many places. This function is called the natural exponential function. However, for most people, this is simply the exponential. The second thing is here’s an opportunity to use the constant multiple rule. So when I take the derivative, I’m differentiating 5 times 2 to the x. And 2 to the x is an exponential function, but 5 is just a constant. By the constant multiple rule, I can pull that out. And then I’m left with this derivative of a simple exponential function. Christian Parkinson GRE Prep: Calculus I Practice Problem Solutions 3 so fis constant. Let f(x) = x2+sin(x) for x0. The temptation here is to use the power rule or the exponential rule but in the. The function (f(x)=e^x ) is the only exponential function (b^x ) with tangent line at (x=0 ) that has a slope of 1. As we see later in the text, having this property makes the natural exponential function the most simple exponential function to use in many instances. Math AP®︎/College Calculus AB Limits and continuity Determining limits using algebraic properties of limits: limit properties Limits of composite functions AP.CALC: LIM‑1 (EU), LIM‑1.D (LO), LIM‑1.D.1 (EK), LIM‑1.D.2 (EK).

1.4 Limits Of Exponential Functions Ap Calculus 14th Edition

Multiply by $$frac{10}{10}$$

$$ begin{align*} displaystylelim_{xto0},frac x {e^{10x} - 1} % & = displaystylelim_{xto0}left(% frac{blue{10}}{blue{10}}cdot frac x {e^{10x} - 1} right)[6pt] % & = displaystylelim_{xto0}left(% frac{1}{blue{10}}cdot frac{blue{10}x}{e^{10x} - 1} right)[6pt] % & = frac{1}{blue{10}}cdot lim_{xto0},frac{blue{10}x}{e^{10x} - 1} end{align*} $$

Step 2Exponential

Rewrite the function as its reciprocal raised to the $$-1$$ power.

$$ frac{1}{10}cdot displaystylelim_{xto0},frac{10x}{e^{10x} - 1} % = frac{1}{10}cdot displaystylelim_{xto0}left(% frac{e^{10x} - 1}{10x} right)^{-1} $$

Step 3

1.4 Limits Of Exponential Functions Ap Calculus Algebra

Exponential

Pass the limit inside the exponent and evaluate (see the page on Limit Laws).

$$ frac{1}{10}cdot displaystylelim_{xto0}left(% blue{frac{e^{10x} - 1}{10x}} right)^{-1} % = frac{1}{10}left(% blue{displaystylelim_{xto0},frac{e^{10x} - 1}{10x}} right)^{-1} % = frac 1 {10} (blue 1)^{-1} % = frac 1 {10} $$

Answer

1.4 Limits Of Exponential Functionsap Calculus Calculator

1.4 limits of exponential functionsap calculus equation

1.4 Limits Of Exponential Functions Ap Calculus Frq

$$ displaystyle lim_{xto0}, frac x {e^{10x} - 1} = frac 1 {10} $$