22 Section 5.1 Infinity, Limits, and Power Functions
a. [latex]f(x)=\frac{-x^{2}}{24}[/latex]
b. [latex]g(x)=\frac{x}{4}+5[/latex]
c. [latex]h(x)=-2^x[/latex]
d. [latex]j(x)=2^{-x}[/latex]
e. [latex]k(x)=\frac{1}{3x^{5}}[/latex]
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Answers
a. Yes, this is a power function. [latex]k=-\frac{1}{24}[/latex] and [latex]p=2[/latex].
b. No, this is not a power function. Power functions must follow the form of [latex]y=kx^p[/latex], and this function has [latex]+5[/latex] on the end.
c. No, this is not a power function. With the [latex]x[/latex] in the exponent, this is an exponential function. A power function has a base [latex]x[/latex].
d. No, this is not a power function. The negative with the [latex]x[/latex] in the exponent does not change that this is an exponential function.
e. Yes, this is a power function. [latex]k=\frac{1}{3}[/latex] and [latex]p=-5[/latex].
2. Determine the value of each limit, if possible.
a. [latex]\displaystyle\lim_{x \to \infty} -\frac{1}{x}[/latex]
b. [latex]\displaystyle\lim_{x \to -\infty} -200x+500[/latex]
c. [latex]\displaystyle\lim_{x \to \infty} -2^x[/latex]
d. [latex]\displaystyle\lim_{x \to \infty} \frac{1}{x^4}+1[/latex]
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Answers
a. This limit is 0. When the numerator of a fraction does not change, and the denominator gets increasingly bigger, the fraction will approach 0.
b. As this is a linear function with negative slope, when the x value gets smaller and smaller, this function will approach [latex]\infty[/latex].
c. The function is an exponential growth function that has been reflected over the [latex]x[/latex]-axis. This means that the limit will be [latex]-\infty[/latex].
d. The fraction will get smaller and smaller as [latex]x[/latex] gets larger, so the fraction is approaching 0, but with the +1, the limit of this will approach 1.