Math Problem Statement

Evaluate the following limits: 1) lim (x→-1) e^x, 2) lim (x→0) 2^(x-1), 3) lim (x→1) ln(x^2 + x - 1), 4) lim (x→π/4) sin(π/4 - x), 5) lim (x→-π/2) cos(x + π/2).

Solution

From the image you uploaded, there are five limits to evaluate. Here are the solutions to each:

  1. limx1ex\lim_{x \to -1} e^x

    • The exponential function is continuous, and therefore, the limit as xx approaches any real number is simply the function evaluated at that number. So: limx1ex=e1=1e\lim_{x \to -1} e^x = e^{-1} = \frac{1}{e}
  2. limx02x1\lim_{x \to 0} 2^{x-1}

    • The function 2x12^{x-1} is a transformation of the function 2x2^x, shifted down by 1. Since the exponential function 2x2^x is continuous, we directly substitute x=0x = 0: limx02x1=201=21=12\lim_{x \to 0} 2^{x-1} = 2^{0-1} = 2^{-1} = \frac{1}{2}
  3. limx1ln(x2+x1)\lim_{x \to 1} \ln(x^2 + x - 1)

    • First, substitute x=1x = 1 into the function: x2+x1=12+11=1x^2 + x - 1 = 1^2 + 1 - 1 = 1
    • The natural log function is continuous where it is defined (i.e., ln(x)\ln(x) for x>0x > 0): limx1ln(x2+x1)=ln(1)=0\lim_{x \to 1} \ln(x^2 + x - 1) = \ln(1) = 0
  4. limxπ4sin(π4x)\lim_{x \to \frac{\pi}{4}} \sin\left(\frac{\pi}{4} - x\right)

    • Since sin\sin is a continuous function: limxπ4sin(π4x)=sin(π4π4)=sin(0)=0\lim_{x \to \frac{\pi}{4}} \sin\left(\frac{\pi}{4} - x\right) = \sin\left(\frac{\pi}{4} - \frac{\pi}{4}\right) = \sin(0) = 0
  5. limxπ2cos(x+π2)\lim_{x \to -\frac{\pi}{2}} \cos\left(x + \frac{\pi}{2}\right)

    • This can be simplified by recognizing a trigonometric identity: cos(x+π2)=sin(x)\cos\left(x + \frac{\pi}{2}\right) = -\sin(x)
    • Therefore: limxπ2cos(x+π2)=sin(π2)=(1)=1\lim_{x \to -\frac{\pi}{2}} \cos\left(x + \frac{\pi}{2}\right) = -\sin\left(-\frac{\pi}{2}\right) = -(-1) = 1

Would you like more details or have any further questions about these solutions? Here are some follow-up points to consider:

  1. What would be the impact on these limits if the approaching values were changed?
  2. How do these specific functions behave around the limit points?
  3. Can these solutions be verified using graphical approaches?
  4. What happens to these functions outside the limits provided?
  5. How can the concept of limits be used in real-world applications?

Tip: When evaluating limits, especially involving trigonometric functions, remember to check for identities that can simplify calculations.

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Math Problem Analysis

Mathematical Concepts

Limits
Exponential functions
Logarithmic functions
Trigonometric functions

Formulas

lim x→a f(x) = f(a) for continuous functions
lim x→0 sin(x)/x = 1
logarithmic identities for ln(x)
trigonometric identities

Theorems

Limit continuity theorem
Squeeze theorem (for sin and cos limits)
Basic properties of logarithms

Suitable Grade Level

Undergraduate Calculus